Business stats assignment

BUSINESS STATISTICS Assignment Submitted to Dr.R Venkata Muni Reddy 9/24/2009

Alliance Business School

Submitted By

GROUP 2:

Abhishek Modi (09PG266)

Ayushee Singh (09PG251)

Kanika Setia (09PG261)

Paras Kumar (09PG271)

Rohit Tomar (09PG281)

Sugandha Huria (09PG291)

Business Statistics September 24, 2009

Alliance Business School, Bangalore Page 2

INTRODUCTION AND BASIC CONCEPTS

Statistics is a mathematical science pertaining to the collection, analysis, interpretation or

explanation, and presentation of data. Statisticians improve the quality of data with the design of

experiments and survey sampling. Statistics also provides tools for prediction and forecasting

using data and statistical models. Statistics is applicable to a wide variety of academic

disciplines, including natural and social sciences, government, and business.

Statistical methods can be used to summarize or describe a collection of data; this is called

descriptive statistics. This is useful in research, when communicating the results of experiments.

In addition, patterns in the data may be modeled in a way that accounts for randomness and

uncertainty in the observations, and are then used to draw inferences about the process or

population being studied; this is called inferential statistics. Inference is a vital element of

scientific advance, since it provides a prediction (based in data) for where a theory logically

leads. To further prove the guiding theory, these predictions are tested as well, as part of the

scientific method. If the inference holds true, then the descriptive statistics of the new data

increase the soundness of that hypothesis. Descriptive statistics and inferential statistics (a.k.a.,

predictive statistics) together comprise applied statistics.

There is also a discipline called mathematical statistics, which is concerned with the theoretical

basis of the subject

Statistics is about gaining information from sets of data. Sometimes you want to represent a lot

of complicated information from a large data set in a way that is easily understood. This is called

descriptive statistics.

Statistics is about gaining information from sets of data. Sometimes you want to represent a lot

of complicated information from a large data set in a way that is easily understood. This is called

descriptive statistics.

An example of this is the so-called worm plot used in cricket: over the cause of a cricket match

there can be many hundreds of balls and runs. In the worm plot depicted below, England‘s

performance is described by the blue line and the West Indies‘ by the green line. You can see at a



glance that, although there was variation in the run rate, England consistently scored at a higher

rate than the West Indies, and so won the match.

Although some information has been lost - you don't know for instance from which balls in the

overs runs were taken - this summary graph clearly displays all of the meaningful information.

The human mind is very visual, and this is why graphics, such as graphs or pie charts, are very

good for conveying statistical information.

The other branch of statistics is called inference statistics. This used to obtain information about

a large set of data from a smaller sample. Think of opinion polls. Here, the statistician randomly

selects a group of people, a thousand say, and asks them about their opinion, for example

whether or not they like the current government. It is then assumed that the opinion of the sample

reflects the opinions of people as a whole.

To be able to do statistics, you first have to learn how to collect, handle and represent data.

USE OF STATISTICAL DATA

Probability theory

Statistics is intimately linked to probability theory. You can use statistics to work out the

probability, the chance that a certain event will occur: if you want to know the chance that your

holiday plane will crash, you think of how many planes usually crash within a year. Since this

number is very small, you deduce that the chance of your plane crashing is small also. You've

done a very simple statistical analysis of the data concerning plane crashes and used it to work

out a probability.

But things also work the other way around: you can use abstract probabilities to help you with

your stats. Say for example you want to test whether a die that is used in a casino is fair. To do



this, you throw the die a great number of times and record the outcomes. You then reason like

this: if the die is fair, then each number should be equally likely. There are six numbers, so each

number should come up in 1/6 of the cases. If this is the case, you decide that it is fair.

This example shows how the abstract theory of probability can help to evaluate real-life

statistics. And this is why probability theory belongs to the basic tool set of a statistician.

So who, apart from opinion pollsters and professional gamblers, uses statistics? Here are a few

examples:

Medicine

Stats and probability theory are absolutely essential in medicine as they are used to test new

drugs and work out the chance that patients develop side effects from the drugs. Tests are

performed on large groups of animals or people and stats are the tool needed to evaluate the tests.

It's essential to get it right, for obvious reasons. Even doctors and nurses who don't perform the

tests themselves need to be well-versed in stats to understand the results and advise their patients

accurately.

Stats and probability theory are also used to assess the risk from things like tobacco and alcohol,

and to see how a certain gene affects people. How likely is it that a person with that gene

develops a certain illness or characteristic?

Medical research cannot do without statistics.

Social and natural sciences

On the face of it, sciences like psychology, sociology or biology do not seem to have much to do

with maths or stats. But all of these have one thing in common: they are based on observations of

the world around us. A psychologist might want to observe people with a certain mental illness, a

biologist the behavior of a virus, and a sociologist a possible link between criminality and drug

abuse. To evaluate these observations, scientists need descriptive statistics. They need to know

how to best collect data and how to represent them in a meaningful way. To interpret the data,

they need inference statistics.



The financial world

A very important thing in the financial world is risk assessment: what is the probability, or risk,

of a company going bankrupt, or the interest rates going up? What is the risk of investing in a

company, or of taking on a mortgage? The insurance industry is based on the idea of risk: the

chance of your house burning down is quite small, but if it does happen, you lose everything.

The insurance company exactly balances the risk of fire with the cost of a fire. They decide what

premium to charge you, so that they still make a profit even though they sometimes pay out huge

amounts.

A good understanding of risk and how it can be described using statistics and probability, is

essential for anyone working in the financial world. Employers in this area often value

mathematicians and statisticians just as highly as people with an economics background.

Politics

Politics is very much about strategy. How should an election campaign be fought? How should a

government deal with other powers? How much money should the health service receive? To

find a good strategy, politicians need to understand public opinion, know about the structure of

society and assess risks. The government employs many statisticians to help them with this. They

can conduct and evaluate a census, and work out the risk of there being an epidemic, or of the

world economy plunging.

During the cold war, game theory, which is closely related to probability theory, was used to

decide whether the US strategy - arming itself to the teeth to deter an attack from the USSR -

was effective.

Reliability theory in manufacturing

When you produce a product, be it a car or a light bulb, you want to know how reliable it is. To

find out, you take a sample of your light bulbs or cars and test them. Just as in an opinion poll,

you can use statistical methods to gain information about the quality of your product from this

sample. Reliability theory has become a very important branch within statistics.



Law

Statistics is often used in the courts. Say a DNA sample has been taken from a crime scene. What

is the chance that a defendant matches this DNA even though he or she is innocent?

In fact, the use of statistics in court can be very tricky, because people are easily confused by it.

A few years ago, a woman called Sally Clark was jailed for the murder of her two children. She

said that they both died of cot death, but the jury was told by an "expert witness" that the

probability of two children dying of cot death in the same family is extremely low, so they

decided that she must have killed them. But this reasoning is flawed. This was recognized later

and the woman was eventually released.

All of us

Everyday life is full of statistics that we need to understand. Politicians and commercial

organizations use stats to convince us to vote for them or buy their products. We need stats to

understand the risks involved in taking a certain medicine or making financial decisions. A basic

grasp of statistics means that you don't have to rely on someone else to make up your mind about

these things. You don‘t need to be an expert — a little basic knowledge can go a long way in

understanding the numbers you are bombarded with every day.

Business Statistics:

The word ‗statistics‘ is derived from the Latin word ‗status‘ meaning a political state. In those

days, therefore, statistics was simply the collection of numerical data by the state or kings. Now,

statistics is the scientific method of analyzing quantitative information. It includes methods of

collection, classification, description and interpretation of data. It simply refers to numerical

description of the quantitative aspects of a phenomenon.

Definition: By statistics we mean aggregate of facts affected to a marked extent by multiplicity

of causes numerically expressed, enumerated or estimated according to reasonable standards of

accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to

each other. Or it can also be defines as a science of collection, presentation, analysis and

interpretation of numerical data.

Raw Data Raw data represent numbers and facts in the original format in which data have been collected

Example for raw data: Z



Frequency Distribution

Frequency Distribution is a summarized table in which raw data are arranged into classes and

frequencies. It is called grouped data. The grouped data can be classified into two. They are

discrete data and continuous data.

Discrete data Discrete data can take only certain specific values that are whole numbers. Example: Number of

classrooms in a school, number of students in a class. Discrete numbers cannot take fractional

values.

Continuous data

Continuous data can take any numerical value within a specific interval e.g. height in

centimeters; weight in kilograms; income in rupees.

Sources of Data

There are two basic sources of collecting the data. They are (i) Primary source and (ii) Secondary

source. If the data are collected from primary source, it is called primary data. The data collected

from the secondary sources are called the secondary data.

Primary data

Data collected for the first time for a specific purpose is called primary data. They are original in

character. They are collected by individuals or institutions or government for research purpose or

policy decisions. Example: Data collected in a population census by the office of the census

commissioner.

Secondary Data

These data are not originally collected. They are obtained from published or unpublished

sources. Published sources are reports and official publications like annual reports of the bank,

population census, Economic survey of India; unpublished sources are the Government records,

studies made by research institutions. Example for the secondary data: Census data used by

research scholars. The census data are primary to the office of the census commissioner who

collected it and for others it is a secondary data.

Classification of Data

Classification is the process of arranging the collected data into classes and to subclasses

according to their common characteristics. Classification is the grouping of related facts into

classes. E.g. sorting of letters in post office



Types of classification

There are four types of classification. They are

(i) Geographical classification

(ii) Chronological classification

(iii) Qualitative classification

(iv) Quantitative classification

Types of Data

Two types of data: qualitative and quantitative. The way we typically define them, we call data

'quantitative' if it is in numerical form and 'qualitative' if it is not. Notice that qualitative data

could be much more than just words or text. Photographs, videos, sound recordings and so on,

can be considered qualitative data.

The quantitative types argue that their data is 'hard', 'rigorous', 'credible', and 'scientific'. The

qualitative proponents counter that their data is 'sensitive', 'nuanced', 'detailed', and 'contextual'.

The fact that qualitative and quantitative data are intimately related to each other. All

quantitative data is based upon qualitative judgments; and all qualitative data can be

described and manipulated numerically. For instance, think about a very common quantitative

measure in social research -- a self esteem scale. The researchers who develop such instruments

had to make countless judgments in constructing them: how to define self esteem; how to

distinguish it from other related concepts; how to word potential scale items; how to make sure

the items would be understandable to the intended respondents; what kinds of contexts it could

be used in; what kinds of cultural and language constraints might be present; and on and on. The

researcher who decides to use such a scale in their study has to make another set of judgments:

how well does the scale measure the intended concept; how reliable or consistent is it; how

appropriate is it for the research context and intended respondents; and on and on. Believe it or

not, even the respondents make many judgments when filling out such a scale: what is meant by



various terms and phrases; why is the researcher giving this scale to them; how much energy and

effort do they want to expend to complete it, and so on. Even the consumers and readers of the

research will make lots of judgments about the self esteem measure and its appropriateness in

that research context. What may look like a simple, straightforward, cut-and-dried quantitative

measure is actually based on lots of qualitative judgments made by lots of different people.

Quantitative and qualitative data are two types of data.

Qualitative data Qualitative - or categorical measurement expressed not in terms of numbers, but

rather by means of a natural language description. In statistics it is often used interchangeably

with "categorical" data.

For example: favourite colour = "blue"

height = "tall"

Although we may have categories, the categories may have a structure to them. When there is

not a natural ordering of the categories, we call these nominal categories. Examples might be

gender, race, religion, or sport.

When the categories may be ordered, these are called ordinal variables. Categorical variables

that judge size (small, medium, large, etc.) are ordinal variables. Attitudes (strongly disagree,

disagree, neutral, agree, strongly agree) are also ordinal variables, however we may not know

which value is the best or worst of these issues. Note that the distance between these categories is

not something we can measure.

Quantitative data

Quantitative -- or numerical measurement expressed not by means of a natural language

description, but rather in terms of numbers. However, not all numbers are continuous and

measurable -- for example social security number -- even though it is a number it is not

something that one can add or subtract.

For example: favourite colour = "450 nm"

height = "1.8 m"

http://wiktionary.org/wiki/Quantitative

http://en.wikipedia.org/wiki/Blue



Quantitative data always are associated with a scale measure.

Probably the most common scale type is the ratio-scale. Observations of this type are on a scale

that has a meaningful zero value but also have an equidistant measure (i.e., the difference

between 10 and 20 is the same as the difference between 100 and 110). For example a 10 year-

old girl is twice as old as a 5 year-old girl. Since you can measure zero years, time is a ratio-scale

variable. Money is another common ratio-scale quantitative measure. Observations that you

count are usually ratio-scale (e.g., number of widgets).

A more general quantitative measure is the interval scale. Interval scales also have a equidistant

measure. However, the doubling principle breaks down in this scale. A temperature of 50

degrees Celsius is not "half as hot" as a temperature of 100, but a difference of 10 degrees

indicates the same difference in temperature anywhere along the scale. The Kelvin temperature

scale, however, constitutes a ratio scale because on the Kelvin scale zero indicates absolute zero

in temperature, the complete absence of heat. So one can say, for example, that 200 degrees

Kelvin is twice as hot as 100 degrees Kelvin.



GRAPHICAL REPRESENTAION OF DATA













MEASURES OF CENTRAL TENDENCY

In analyzing, statistical data, it is often useful to have numbers describe the complete set of data.

―Measures of central tendency‖ are used because they represent centralized or middle values of

the data. These measures of central tendency are called the ―mean,‖ ―median,‖ and ―mode.‖

The ―mean‖ is a number that represents an ―average‖ of a set of data. It is found by adding the

elements in the set and then dividing that sum by the number of elements in the set.

Definition: The ―mean‖ of a set of data is the sum of the elements in the set divided by the

number of elements in the set.

Figure 1.1: Mean



Analysis of Mean in this example:

The mean temperature is greater than all of the daily temperatures except one, . Thus, is

not a very good representation of the average of the set of data? Extremely high or low values,

such as , affect the mean.

It must be pointed out that because the temperature is not a good representation of the set of

data, possibly the mean is not always the best average. This opens the door to introducing the

―median‖ and ―mode‖ as averages that sometimes are better representations.



The mode is the number that occurs most often in a set of data

A set of data may have more than one mode. For example, in {2, 3, 3, 4, 6, 6}, 3 and 6 are

both modes for the set of data. For a set in which there are two modes, it is sometimes said

to be bimodal, a set of three modes, trimodal, and so on. If no number occurs more often

than the other numbers, then a set of data has

no mode.



Median: The ―median‖ is the middle number of a set of data when the numbers are arranged in

increasing order.

2nd Example for MEDIAN:

If a set of data contains an even number of elements, the median is the value halfway

between the two middle elements. In other words, when there are an even number of

elements in a set of data, the median is found by determining the mean of the two middle

elements.





The Variance:

The variance of a set of observations is the average squared deviation of the data points from

their mean .This measure of dispersion reflects the values of all the measurements.

Solution of the above example :



Standard Deviation: A set of observations is the (positive) square root of the variance of set. the

standard deviation for a normal distribution is the distance along the horizontal axis between the

mean and either horizontal coordinate for which the curve changes concavity.







Example



Example



Probability

Basic concepts:

Independent Events

Two events are independent if the following are true:

* P(A|B) = P(A)P(A|B) = P(A)

* P(B|A) = P(B)P(B|A) = P(B)

* P(A AND B) = P(A) ⋅ P(B)P(A AND B) = P(A) ⋅ P(B)

If AA and BB are independent, then the chance of AA occurring does not affect the chance of

BB occurring and vice versa. For example, two roles of a fair die are independent events. The

outcome of the first roll does not change the probability for the outcome of the second roll. To

show two events are independent, you must show only one of the above conditions. If two events

are NOT independent, then we say that they are dependent.

Sampling may be done with replacement or without replacement.

* With replacement: If each member of a population is replaced after it is picked, then that

member has the possibility of being chosen more than once. When sampling is done with

replacement, then events are considered to be independent, meaning the result of the first pick

will not change the probabilities for the second pick.

* Without replacement: When sampling is done without replacement, then each member of a

population may be chosen only once. In this case, the probabilities for the second pick are

affected by the result of the first pick. The events are considered to be dependent or not

independent.

If it is not known whether AA and BB are independent or dependent, assume they are dependent

until you can show otherwise.

Mutually Exclusive Events

AA and BB are mutually exclusive events if they cannot occur at the same time. This means that

AA and BB do not share any outcomes and P(A AND B) =0.



For example:

Suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A

= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}C = {7,

9}. A AND B= {4,5}A AND B ={4, 5}. P(A AND B) =P(A AND B) = 2/10 and is not equal to

0. Therefore, AA and BB are not mutually exclusive. AA and CC do not have any numbers in

common so P (A AND C) = 0. Therefore, AA and CC are mutually exclusive. If it is not known

whether A and B are mutually exclusive, assume they are not until you can show otherwise.

No overlap in Venn diagram

Equally Likely Cases:

The out comes are said to be equally likely or equally probable if none of them is expected to

occur in preference to other. That is the probability of the occurrence of all the probable events is

threw same and equal. Thus 1) in tossing of a coin, if all the out comes, viz., {H, T} are equally

likely then the coin is unbiased. The probability of getting a head or a tail in tossing a coin is

both ½.

The probability that the experiment results in a successful outcome (S) is:

P(S) = (Number of successful outcomes) / (Total number of equally likely outcomes ) = r / n

Consider the following experiment. An urn has 10 marbles. Two marbles are red, three are green,

and five are blue. If an experimenter randomly selects 1 marble from the urn, what is the

probability that it will be green?

In this experiment, there are 10 equally likely outcomes, three of which are green marbles.

Therefore, the probability of choosing a green marble is 3/10 or 0.30.



Example:

A coin is tossed three times. What is the probability that it lands on heads exactly one time?

The correct answer is (D). If you toss a coin three times, there are a total of eight possible

outcomes. They are: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT. Of the eight possible

outcomes, three have exactly one head. They are: HTT, THT, and TTH. Therefore, the

probability that three flips of a coin will produce exactly one head is 3/8 or 0.375.

Union and intersection

The probability that Events A or B occur is the probability of the union of A and B. The

probability of the union of Events A and B is denoted by P(A ∪ B) .

The probability that Events A and B both occur is the probability of the intersection of A and B.

The probability of the intersection of Events A and B is denoted by P(A ∩ B). If Events A and B

are mutually exclusive, P(A ∩ B) = 0.

Additive and multiplication Rule

Rule of Multiplication:

The probability that Events A and B both occur is equal to the probability that Event A occurs

times the probability that Event B occurs, given that A has occurred.

P (A ∩ B) = P (A) P (B|A)

Example:

An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement

from the urn. What is the probability that both of the marbles are black?

Let A = the event that the first marble is black; and let B = the event that the second marble is

black. We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P (A) =

4/10.

After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P

(B|A) = 3/9.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)

P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15



Rule of Addition:

The probability that Event A and/or Event B occur is equal to the probability that Event A occurs

plus the probability that Event B occurs minus the probability that both Events A and B occur.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B))

Example

A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b)

a work of non-fiction is 0.30, , and (c) both fiction and non-fiction is 0.20. What is the

probability that the student checks out a work of fiction, non-fiction, or both?

Solution: Let F = the event that the student checks out fiction; and let N = the event that the

student checks out non-fiction. Then, based on the rule of addition:

P(F ∪ N) = P(F) + P(N) - P(F ∩ N)

P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50

Conditional probability

The probability that Event A occurs, given that Event B has occurred, is called a conditional

probability. The conditional probability of Event A, given Event B, is denoted by the symbol P

(A|B).

In addition one de_nes the conditional probability P (AB) (read P of A given B) as

P (AB) = P (A\B)

Example

A six-sided die is thrown. What is the probability that the number thrown is prime, given that it

is odd.

The probability of obtaining an odd number is 3/6 = ½. Of these odd numbers, 2 of them are

prime (3 and 5).

P(prime | odd)=P(prime and odd) =2/6=2/3.



RANDOM VARIABLES AND

PROBABILITY DISTRIBUTION:

What is a random variable? In many experiments the outcomes of the experiment can be

assigned numerical values. For instance, if you roll a die, each outcome has a value from 1

through 6. If you ascertain the midterm test score of a student in your class, the outcome is again

a number. A random variable is just a rule that assigns a number to each outcome of an

experiment. These numbers are called the values of the random variable. We often use letters

like X, Y and Z to denote a random variable.

Here are some examples

Examples

1. Experiment: Select a mutual fund; X = the number of companies in the fund portfolio.

The values of X are 2, 3, 4, ...

2. Experiment: Select a soccer player; Y = the number of goals the player has scored

during the season.

The values of Y are 0, 1, 2, 3, ...

3. Experiment: Survey a group of 10 soccer players; Z = the average number of goals

scored by the players during the season.

The values of Z are 0, 0.1, 0.2, 0.3, ...., 1.0, 1.1, ...

Types of random variable:

Discrete and Continuous Random Variables

A discrete random variable can take on only specific, isolated numerical values, like the

outcome of a roll of a die, or the number of dollars in a randomly chosen bank account. Discrete

random variables that can take on only finitely many values (like the outcome of a roll of a die)

are called finite random variables. Discrete random variables that can take on an unlimited

number of values (like the number of stars estimated to be in the universe) are infinite discrete

random variables.

A continuous random variable, on the other hand, can take on any values within a continuous

range or an interval, like the temperature in Central Park, or the height of an athlete in

centimeters.



Examples

Random Variable Values Type

Flip a coin three times; X =

the total number of heads.

{0, 1, 2, 3} Finite

There are only four possible

values for X.

Select a mutual fund; X = the

number of companies in the

fund portfolio.

{2, 3, 4, ...} Discrete Infinite

There is no stated upper limit

to the size of the portfolio.

Measure the length of an

object; X = its length in

centimeters.

Any positive real number Continuous

The set of possible

measurements can take on

any positive value.

Discrete Probability Distributions

If a random variable is a discrete variable, its probability distribution is called a discrete

probability distribution.

An example will make this clear. Suppose you flip a coin two times. This simple statistical

experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random

variable X represent the number of Heads that result from this experiment. The random variable

X can only take on the values 0, 1, or 2, so it is a discrete random variable.

The probability distribution for this statistical experiment appears below.

Number of heads Probability

0 0.25

1 0.50

2 0.25

The above table represents a discrete probability distribution because it relates each value of a

discrete random variable with its probability of occurrence. In subsequent lessons, we will cover

the following discrete probability distributions.

Binomial probability distribution

Hypergeometric probability distribution

Multinomial probability distribution

Poisson probability distribution

http://stattrek.com/Help/Glossary.aspx?Target=Probability_distribution

http://stattrek.com/Help/Glossary.aspx?Target=Statistical_experiment



http://stattrek.com/Lesson2/Binomial.aspx

http://stattrek.com/Lesson2/Hypergeometric.aspx

http://stattrek.com/Lesson2/Multinomial.aspx



Binomial Probability

The binomial probability refers to the probability that a binomial experiment results in

exactly x successes. For example, in the above table, we see that the binomial probability

of getting exactly one head in two coin flips is 0.50.

Given x, n, and P, we can compute the binomial probability based on the following

formula:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x

successes. If the probability of success on an individual trial is P, then the binomial

probability is:

b(x; n, P) = nCx * Px * (1 - P)

n - x

Example 1

Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the

number of successes is equal to 2, and the probability of success on a single trial is 1/6 or

about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = 5C2 * (0.167)2 * (0.833)

3

b(2; 5, 0.167) = 0.161

Cumulative Binomial Probability

A cumulative binomial probability refers to the probability that the binomial random variable

falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or

equal to a stated upper limit).

For example, we might be interested in the cumulative binomial probability of obtaining 45 or

fewer heads in 100 tosses of a coin (see Example 1 below). This would be the sum of all these

individual binomial probabilities.

b(x < 45; 100, 0.5) =

b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + ... + b(x = 44; 100, 0.5) + b(x = 45; 100, 0.5)



Example 1

What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?

Solution: To solve this problem, we compute 46 individual probabilities, using the binomial

formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 45; 100, 0.5)

b(x < 45; 100, 0.5) = 0.184

Example 2

The probability that a student is accepted to a prestigeous college is 0.3. If 5 students from the

same school apply, what is the probability that at most 2 are accepted?

Solution: To solve this problem, we compute 3 individual probabilities, using the binomial

formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3)

b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087

b(x < 2; 5, 0.3) = 0.8369

Hypergeometric Distribution

A hypergeometric random variable is the number of successes that result from a

hypergeometric experiment. The probability distribution of a hypergeometric random variable is

called a hypergeometric distribution.

Given x, N, n, and k, we can compute the hypergeometric probability based on the following

formula:

Hypergeometric Formula. Suppose a population consists of N items, k of which are successes.

And a random sample drawn from that population consists on n items, x of which are successes.

Then the hypergeometric probability is:

h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]



The hypergeometric distribution has the following properties:

The mean of the distribution is equal to n * k / N .

The variance is n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ] .

Example 1

Suppose we randomly select 5 cards without replacement from an ordinary deck of playing

cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)?

Solution: This is a hypergeometric experiment in which we know the following:

N = 52; since there are 52 cards in a deck.

k = 26; since there are 26 red cards in a deck.

n = 5; since we randomly select 5 cards from the deck.

x = 2; since 2 of the cards we select are red.

We plug these values into the hypergeometric formula as follows:

h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]

h(2; 52, 5, 26) = [ 26C2 ] [ 26C3 ] / [ 52C5 ]

h(2; 52, 5, 26) = [ 325 ] [ 2600 ] / [ 2,598,960 ] = 0.32513

Thus, the probability of randomly selecting 2 red cards is 0.32513.

Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment.

The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the

Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of

successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e-μ

) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately

equal to 2.71828.

http://stattrek.com/Help/Glossary.aspx?Target=Variance

http://stattrek.com/Help/Glossary.aspx?Target=Probability_distribution



The Poisson distribution has the following properties:

The mean of the distribution is equal to μ .

The variance is also equal to μ .

Example 1

The average number of homes sold by the Acme Realty company is 2 homes per day. What is

the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

μ = 2; since 2 homes are sold per day, on average.

x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.

e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e-μ

) (μx) / x!

P(3; 2) = (2.71828-2

) (23) / 3!

P(3; 2) = (0.13534) (8) / 6

P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

Continuous probability distribution

a probability distribution is called continuous if its cumulative distribution function is

continuous, which means that it belongs to a random variable X for which Pr[ X = x ] = 0 for all x

in R.

Another convention reserves the term continuous probability distribution for absolutely

continuous distributions. These distributions can be characterized by a probability density

function: a non-negative Lebesgue integrable function f defined on the real numbers such that

Discrete distributions and some continuous distributions (like the devil's staircase) do not admit

such a density.

http://stattrek.com/Help/Glossary.aspx?Target=Variance

http://en.wikipedia.org/wiki/Absolute_continuity



http://en.wikipedia.org/wiki/Probability_density_function



http://en.wikipedia.org/wiki/Lebesgue_integration

http://en.wikipedia.org/wiki/Singular_function



The Normal Equation

Normal distributions model (some) continuous random variables. Strictly, a Normal random

variable should be capable of assuming any value on the real line, though this requirement is

often waived in practice. For example, height at a given age for a given gender in a given racial

group is adequately described by a Normal random variable even though heights must be

positive.

A continuous random variable X, taking all real values in the range is said to follow a

Normal distribution with parameters µ and if it has probability density function

This probability density function (p.d.f.) is a symmetrical, bell-shaped curve, centred at its

expected value µ. The variance is .

Many distributions arising in practice can be approximated by a Normal distribution. Other

random variables may be transformed to normality.

The simplest case of the normal distribution, known as the Standard Normal Distribution, has

expected value zero and variance one. This is written as N(0,1).

Examples

Example 1

An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard

deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that

an Acme light bulb will last at most 365 days?

http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#contvar

http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#pdf

http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#expval

http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#variance



Solution: Given a mean score of 300 days and a standard deviation of 50 days, we want to find

the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the

following:

The value of the normal random variable is 365 days.

The mean is equal to 300 days.

The standard deviation is equal to 50 days.

We enter these values into the Normal Distribution Calculator and compute the cumulative

probability. The answer is: P( X < 365) = 0.90. Hence, there is a 90% chance that a light bulb

will burn out within 365 days.

Uniform Distribution

Uniform distributions model (some) continuous random variables and (some) discrete random

variables. The values of a uniform random variable are uniformly distributed over an interval.

For example, if buses arrive at a given bus stop every 15 minutes, and you arrive at the bus stop

at a random time, the time you wait for the next bus to arrive could be described by a uniform

distribution over the interval from 0 to 15.

A discrete random variable X is said to follow a Uniform distribution with parameters a and b,

written X ~ Un(a,b), if it has probability distribution

P(X=x) = 1/(b-a)

where

x = 1, 2, 3, ......., n.

A discrete uniform distribution has equal probability at each of its n values.

A continuous random variable X is said to follow a Uniform distribution with parameters a and

b, written X ~ Un(a,b), if its probability density function is constant within a finite interval [a,b],

and zero outside this interval (with a less than or equal to b).

The Uniform distribution has expected value E(X)=(a+b)/2 and variance {(b-a)2}/12.

Example



SAMPLING AND SAMPLING DISTRIBUTION

Sampling: The process of inferring something about a large group of elements by studying

only a part of it is known as sampling. The collection of all elements about which some reference

is to be made is called the population.

Sampling Distribution: Sampling theory is the study of relationship between a population

and samples drawn from the population, and it is applicable to random sample only. We will

discuss how to estimate the true value of the population parameters (population mean, population

standard deviation and population proportion,…etc) by using sample statistic like sample mean,

sample standard deviation, sample proportion,…etc and to find the limits of accuracy of

estimates based on samples.

The Sampling distribution of a sample statistic calculated from a sample of an

measurements is the probability distribution of a statistic.

Eg : If x has been calculated from a sample of n = 25 measurements selected from a

population with mean µ=0.3 and standard deviation Sigma= 0.005, the sampling

distribution provides information about the behavior (mean) of in repeated sampling.

-if you continue to take samples of data and compute every possible combination of samples (i.e.

all permutations or combinations) of size n then the sample statistics/point estimators can have

their own distribution.

-so each sample statistic/point estimator will have its own distributions with its own mean,

variance, and standard deviation.

-we we know what type of distribution this is we can make probability statements from it and

assess how close the point estimates are to the population parameters (i.e. how close _

x is to μ)

1. Sampling Distribution – the probability distribution of any particular sampling statistic.

2. Law of Large Numbers – if we draw observations from a population with a finite mean

μ at random, as we increase the number of observations we draw the value of the sample mean (_

x ) gets closer and closer to the population mean.



-note that this makes sense b/c as you increase the size of your sample it gets closer to the size of

the population. So it begins to look more and more like the population itself. For this reason the

mean should approach the population mean.

3. Sampling Distribution of _

x - this is the probability distribution of all possible values of

a sample mean given a certain size sample n.

Ex:

-Suppose we have a distribution as follows:

If we want to create a sampling distribution we

would take samples of size n, let‘s say 15

from the distribution to the right and from

each sample obtain a mean, variance,

and standard deviation.

…..continue with process for all possible samples of size 15. If we do this we can take the

values from each sample and create its own distribution as shown below.

Graphically:

-notice now we have a distribution of sample means.

This distribution is created from the means of each

sample and it has its own variance and standard

deviation. Note that they should be much

smaller than the distribution sampled from since we created it from the sample means of the data.

x

15 20 25

Sample 2-

has own_

x

, s2, & s

Sample1-

has own_

x

, s2, & s

_

x

20



4. Characteristics of the Sampling Distribution of _

x

a. E (_

x ) = μ so the mean of all values of _

x should be the population mean μ

b. Standard Deviation of _

x - called the standard error of the mean it tells us how close our

estimates of the mean are to the actual mean.

i. finite population value – σ _

x = )1/()( NnN * (σ / n )

ii. infinite population - σ _

x = σ / n

note: σ = population variance, N = population size, n = sample size; must still use the infinite

population estimate if n/N < 5% of the population size.

Notice what happens to the distribution of p� as the sample size grows larger…

CENTRAL LIMIT THEOREM

When choosing n and it is a SRS we can assume that the sampling distribution of _

x ~N as N gets

larger and larger. If it is greater than 30 we assume it is Normal.

-if the population is normal, then the sampling distribution must be normal and this rule does not

apply. This is for any size of sample.

-as n increases the variance and standard deviation get tighter and there is a higher probability

that the sample means is within a certain distance of the actual population mean.



Given a random sample, Y1, Y2, … , Yn where E[Yi] = μ and Var(Yi) = σ2

→ N(μ, σ2⁄n) as n → ∞

The CLT is approximately true for finite n and the approximation improves as n gets

larger.Sometimes we only need a few observations for it to kick in, sometimes we need more.

This depends on how ―far from normal‖ the data is to begin with.

Example 5.13 Y denotes the number of eye facets in a fruit fly. Clearly, Y is not continuous and

hence, cannot be normally distributed. Figure 5.14 illustrates how the CLT still kicks in.

Point estimation

Definition:

Point estimation (point estimate) is a single numerical value used as an

estimate of a population parameter.

population parameter

Properties of point estimators

Definition: Properties of point estimators are as follows:

• unbiasedness – a property of point estimator that occurs whenever the expected value of the

point estimator is equal to the population parameter it estimates.

• consistency – a property of point estimator that occurs whenever lar ger sample sizes tend to

provide point estimates closer to the population parameter.

• (relative) efficiency – if we have two unbiased point estimators of the same



population parameter, the point estimator with the smaller variance is said to have gr eater

efficiency than the other.

Usually, we do not know the population mean and standard deviation. Our goal is to estimate

these numbers. The standard way to accomplish this is to use the sample mean and standard

deviation as a best guess for the true population mean and standard deviation. We call this "best

guess" a point estimate.

A Point Estimate is a statistic that gives a plausible estimate for the value in question.

Example:

x is a point estimate for

s is a point estimate for

A point estimate is unbiased if its mean represents the value that it is estimating.

Interval Estimates: An interval estimator (or confidence interval) is a formula that tells us how

to use sample data to calculate an interval that estimates a population parameter. An alternative is

that a certain interval contains the true mean.

Interval Estimates Example: Plugging in the values, the confidence interval is

72.34< <77.66 is the 95% confidence interval for

So there is a 95% probability that this interval contains the true mean

Confidence Intervals

We are not only interested in finding the point estimate for the mean, but also determining how

accurate the point estimate is. The Central Limit Theorem plays a key role here. We assume

that the sample standard deviation is close to the population standard deviation (which will

almost always be true for large samples). Then the Central Limit Theorem tells us that the

standard deviation of the sampling distribution is

2.66)(752.66)-(75

.95)16

2.131(5)X

16

2.131(5)-X( P



We will be interested in finding an interval around x such that there is a large probability that the

actual mean falls inside of this interval. This interval is called a confidence interval and the large

probability is called the confidence level.

Example

Suppose that we check for clarity in 50 locations in Lake Tahoe and discover that the average

depth of clarity of the lake is 14 feet. Suppose that we know that the standard deviation for the

entire lake's depth is 2 feet. What can we conclude about the average clarity of the lake with a

95% confidence level?

Solution

We can use x to provide a point estimate for . How accurate is x as a point estimate? We

construct a 95% confidence interval for as follows. We draw the picture and realize that we

need to use the table to find the z-score associated to the probability of .025 (there is .025 to the

left and .025 to the right).

We arrive at z = -1.96. Now we solve for x:

x - 14 x - 14

-1.96 = =

2/ 0.28

Hence

x - 14 = -0.55

We say that 0.55 is the margin of error.

We have that a 95% confidence interval for the mean clarity is

(13.45,14.55)

In other words there is a 95% chance that the mean clarity is between 13.45 and 14.55.



In general if zc is the z value associated with c% then a c% confidence interval for the mean is

Example

1000 randomly selected Americans were asked if they believed the minimum wage should be

raised. 600 said yes. Construct a 95% confidence interval for the proportion of Americans who

believe that the minimum wage should be raised.

Solution:

We have

p = 600/1000 = .6 zc = 1.96 and n = 1000

We calculate:

Hence we can conclude that between 57 and 63 percent of all Americans agree with the proposal.

In other words, with a margin of error of .03 , 60% agree.

Confidence Interval for a Mean When the Population Standard Deviation is

Unknown

When the population is normal or if the sample size is large, then the sampling distribution will

also be normal, but the use of s to replace is not that accurate. The smaller the sample size the

worse the approximation will be. Hence we can expect that some adjustment will be made based

on the sample size. The adjustment we make is that we do not use the normal curve for this

approximation. Instead, we use the Student t distribution that is based on the sample size. We

proceed as before, but we change the table that we use. This distribution looks like the normal

distribution, but as the sample size decreases it spreads out. For large n it nearly matches the

normal curve. We say that the distribution has n - 1 degrees of freedom.



Example

Suppose that we conduct a survey of 19 millionaires to find out what percent of their income the

average millionaire donates to charity. We discover that the mean percent is 15 with a standard

deviation of 5 percent. Find a 95% confidence interval for the mean percent. Assume that the

distribution of all charity percents is approximately normal.

Solution

We use the formula:

(Notice the t instead of the z and s instead of s)

We get

15 tc 5 /

Since n = 19, there are 18 degrees of freedom. Using the table in the back of the book, we have

that

tc = 2.10

Hence the margin of error is

2.10 (5) / = 2.4

We can conclude with 95% confidence that the millionaires donate between

12.6% and 17.4% of their income to charity.



Confidence Intervals for Proportions and

Choosing the Sample Size

A Large Sample Confidence Interval for a Population Proportion

Recall that a confidence interval for a population mean is given by

Confidence Interval for a Population Mean

zc s

x

We can make a similar construction for a confidence interval for a population proportion.

Instead of x, we can use p and instead of s, we use , hence, we can write the

confidence interval for a large sample proportion as

Confidence Interval Margin of Error for a Population Proportion

Example

1000 randomly selected Americans were asked if they believed the minimum wage should be

raised. 600 said yes. Construct a 95% confidence interval for the proportion of Americans who

believe that the minimum wage should be raised.



Solution:

We have

p = 600/1000 = .6 zc = 1.96 and n = 1000

We calculate:

Hence we can conclude that between 57 and 63 percent of all Americans agree with the proposal.

In other words, with a margin of error of .03 , 60% agree.

Calculating n for Estimating a Mean

Example

Suppose that you were interested in the average number of units that students take at a two year

college to get an AA degree. Suppose you wanted to find a 95% confidence interval with a

margin of error of .5 for knowing = 10. How many people should we ask?

Solution

Solving for n in

Margin of Error = E = zc /

we have

E = zc

zc

=

E

Squaring both sides, we get



We use the formula:

Example

A Subaru dealer wants to find out the age of their customers (for advertising purposes). They

want the margin of error to be 3 years old. If they want a 90% confidence interval, how many

people do they need to know about?

Solution:

We have

E = 3, zc = 1.65

but there is no way of finding sigma exactly. They use the following reasoning: most car

customers are between 16 and 68 years old hence the range is

Range = 68 - 16 = 52

The range covers about four standard deviations hence one standard deviation is about

52/4 = 13

We can now calculate n:

Hence the dealer should survey at least 52 people.



Finding n to Estimate a Proportion

Example

Suppose that you are in charge to see if dropping a computer will damage it. You want to find

the proportion of computers that break. If you want a 90% confidence interval for this

proportion, with a margin of error of 4%, How many computers should you drop?

Solution

The formula states that

Squaring both sides, we get that

zc2

p(1 - p)

E2 =

n

Multiplying by n, we get

nE2 = zc

2[p(1 - p)]

This is the formula for finding n.

Since we do not know p, we use .5 ( A conservative estimate)

We round 425.4 up for greater accuracy

We will need to drop at least 426 computers. This could get expensive.



Probability Sampling

A probability sample is one in which each member of the population has an equal chance of

being selected - there are four main types of probability sample. The decision as to which

sample to use is dependent upon the nature of the research aim, the desired level of accuracy in

the sample and the availability of a good sampling frame, money and time.

1. Simple Random Sampling

2. Systematic Sampling

3. Stratified Sampling

4. Cluster Sampling

1) Simple Random Sampling

Put simply, this method is where we select a group of people for a study from a larger group i.e.

from a population. Each individual is chosen randomly by chance, and therefore each person has

the same chance as any other of being selected. The easiest way of selecting a sample using this

method is to first obtain a complete sampling frame. Once this has been achieved, each person

within the frame should be allocated a unique reference number starting at one. The size of the

sample must be decided and then that many numbers should be selected, from the table of

random numbers. If the sampling frame consists of 500 people, three digit numbers must be

selected from the random number table, similarly if the highest identifying number on the

sampling frame is a two digit number e.g. 50 you must select two digit numbers from the random

number table. If, as in the example below, the numbers are five digits, simply decide on any two

digits (e.g. first two or last two) and stick to this for the rest of the procedure.

Example

Random Numbers;

Select numbers from every third column and every row. If a number

comes up twice or is larger than the population number, discard it.

Be sure to stick to the pattern of movement through the table.

87456 34098 88900 11128

87456 34098 88900 64554

45666 77789 82276 12555

22333 45767 87900 99989

2) Systematic Sampling

Systematic sampling is very similar to simple random sampling, except instead of selecting

random numbers from tables, you move through the sample frame picking every nth name.



In order to do this, it is necessary to work out the sampling fraction. This is done by dividing the

population by the desired sample.

Example

For a population of 100,000 and a desired sample of 2,000, the sampling fraction is 2/100 or

1/50. This means that you would select one person out of every fifty in the population. With

this method, with the sampling fraction of 1/50, the starting point must be within the first 50

people in your list.

This method does bring about a problem worth highlighting. If you used a sampling frame

which is arranged by gender or marital status, problems could occur i.e. if the list was arranged;

Husband/Wife/Husband/Wife etc. and if every tenth person was to be interviewed, there would

be an increased chance of males being selected. This is known as periodicity – if this exists in

the frame it is necessary to either mix up the cases or use Simple Random Sampling.

3) Stratified Sampling

Stratified sampling is a modification of Simple Random Sampling and Systematic Sampling and

is designed to produce a more representative and thus more accurate sample. A stratified sample

is obtained by taking samples from each sub-group of a population. These could be, for

example, age, gender or marital status. The rationale here is to choose 'stratification variables'

that have a major influence on the survey results.

For example, in a lifestyle survey 'age' is likely to have a key effect on 'lifestyle' and you might

want to ensure your sample contains the correct proportion of residents from each age group.

Remember, stratification in this way will only be possible when selecting the sample if the (in

this case) age of the resident is known on the sampling frame.

Having selected the variable, such as age or gender, you need to order the sampling frames into

groups according to the category, and then use systematic sampling to select the appropriate

proportion of people within each variable.

4) Cluster Sampling

This technique is perhaps the most economical of those looked at so far, particularly if face-to-

face interviewing is to be used. As its name suggests, it is a combination of several different

samples. The entire population is divided into groups, or clusters and a random sample of these

clusters are selected. Following that, smaller 'clusters' are chosen from within the selected

clusters.

Multistage cluster sampling is often used when a random sample would produce a list of subjects

so widely scattered geographically that surveying them would prove to be far too expensive. It

should, however, be noted that sampling errors are larger when using cluster sampling.



Example

Stage 1: Define population - (say) adults 16+ living in the South East of England.

Stage 2: Select (say) 100 electoral wards from the SE at random

Stage 3: Select a member of smaller areas (e.g. EDS) from within each selected ward.

Stage 4: Interview all residents within the smaller areas (alternatively, select a sample

from the each smaller area.

Non-Probability Sampling

For quantitative surveys, probability sampling should be our preferred approach where possible.

It allows randomness to drive the selection and allows estimates of the accuracy of survey

findings to be obtained. The most likely situation for non-probability sampling to be needed is

when there is either no sampling frame or the population is so widely dispersed that cluster

sampling would be too inefficient. Non-probability techniques are cheaper than probability

sampling, and are often used in exploratory studies e.g. for hypothesis generation. There are five

main non-probability sampling techniques;

1. Purposive Sampling

2. Quota Sampling

3. Convenience Sampling

4. Snowball Sampling

5. Self-Selection

1) Purposive Sampling

Purposive sampling is a method where the participants are selected by the researcher

subjectively. The researcher will pick a sample that he/she believes is representative to the

population of interest. Respondents are not selected randomly but by using the judgment of the

interviewers.

2) Quota Sampling

Quota Sampling is perhaps most commonly used in face-to-face interviewing. Interviewers on

the street are usually looking for a specific type of respondent – age, gender are the most

frequently used 'quota controls'. Quotas are given to interviewers and are organized so that the

final sample is representative of the population. It is impossible to estimate the accuracy of the

sample because it is not random.

3) Convenience Sampling

Similar to quota sampling, convenience sampling is a technique often used in face-to-face

interviewing. A convenience sample is when the interviewer simply stops anyone in the street or

knock on doors asking anyone to participate and interviewing anyone willing to help. It is hard

to draw any meaningful conclusions from the results obtained due to the lack of randomness,

meaning the likelihood of bias is high.



4) Snowball Sampling

This approach is often used when trying to interview hard to reach groups such as unemployed

people or Black or Minority Ethnic residents.. You initially contact a few potential respondents,

interview them and then ask if they know of anybody else with the same characteristics you are

looking for.

5) Self-Selection

This technique is self-explanatory – respondents themselves decide whether to take part in the

survey or not.

INFERENCES BASED ON A SINGLE SAMPLE :TESTS

OF HYPOTHESIS

Hypothesis Test

Setting up and testing hypotheses is an essential part of statistical inference. In order to formulate

such a test, usually some theory has been put forward, either because it is believed to be true or

because it is to be used as a basis for argument, but has not been proved, for example, claiming

that a new drug is better than the current drug for treatment of the same symptoms.

In each problem considered, the question of interest is simplified into two competing claims /

hypotheses between which we have a choice; the null hypothesis, denoted H0, against the

alternative hypothesis, denoted H1. These two competing claims / hypotheses are not however

treated on an equal basis: special consideration is given to the null hypothesis.

We have two common situations:

1. The experiment has been carried out in an attempt to disprove or reject a particular

hypothesis, the null hypothesis, thus we give that one priority so it cannot be rejected

unless the evidence against it is sufficiently strong. For example,

H0: there is no difference in taste between coke and diet coke against

H1: there is a difference.

2. If one of the two hypotheses is 'simpler' we give it priority so that a more 'complicated'

theory is not adopted unless there is sufficient evidence against the simpler one. For

example, it is 'simpler' to claim that there is no difference in flavour between coke and

diet coke than it is to say that there is a difference.



The hypotheses are often statements about population parameters like expected value and

variance; for example H0 might be that the expected value of the height of ten year old boys in

the Scottish population is not different from that of ten year old girls. A hypothesis might also be

a statement about the distributional form of a characteristic of interest, for example that the

height of ten year old boys is normally distributed within the Scottish population.

The outcome of a hypothesis test test is "Reject H0 in favour of H1" or "Do not reject H0".

Null Hypothesis

The null hypothesis, H0, represents a theory that has been put forward, either because it is

believed to be true or because it is to be used as a basis for argument, but has not been proved.

For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is

no better, on average, than the current drug. We would write

H0: there is no difference between the two drugs on average.

We give special consideration to the null hypothesis. This is due to the fact that the null

hypothesis relates to the statement being tested, whereas the alternative hypothesis relates to the

statement to be accepted if / when the null is rejected.

The final conclusion once the test has been carried out is always given in terms of the null

hypothesis. We either "Reject H0 in favour of H1" or "Do not reject H0"; we never conclude

"Reject H1", or even "Accept H1".

If we conclude "Do not reject H0", this does not necessarily mean that the null hypothesis is true,

it only suggests that there is not sufficient evidence against H0 in favor of H1. Rejecting the null

hypothesis then, suggests that the alternative hypothesis may be true.

Alternative Hypothesis

The alternative hypothesis, H1, is a statement of what a statistical hypothesis test is set up to

establish. For example, in a clinical trial of a new drug, the alternative hypothesis might be that

the new drug has a different effect, on average, compared to that of the current drug. We would

write

H1: the two drugs have different effects, on average.

The alternative hypothesis might also be that the new drug is better, on average, than the current

drug. In this case we would write

H1: the new drug is better than the current drug, on average.



The final conclusion once the test has been carried out is always given in terms of the null

hypothesis. We either "Reject H0 in favour of H1" or "Do not reject H0". We never conclude

"Reject H1", or even "Accept H1".

If we conclude "Do not reject H0", this does not necessarily mean that the null hypothesis is true,

it only suggests that there is not sufficient evidence against H0 in favor of H1. Rejecting the null

hypothesis then, suggests that the alternative hypothesis may be true.

Simple Hypothesis

A simple hypothesis is a hypothesis which specifies the population distribution completely.

Examples

1. H0: X ~ Bi(100,1/2), i.e. p is specified

2. H0: X ~ N(5,20), i.e. µ and are specified

Composite Hypothesis

A composite hypothesis is a hypothesis which does not specify the population distribution

completely.

Examples

1. X ~ Bi(100,p) and H1: p > 0.5

2. X ~ N(0, ) and H1: unspecified

Type I Error

In a hypothesis test, a type I error occurs when the null hypothesis is rejected when it is in fact

true; that is, H0 is wrongly rejected.

For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is

no better, on average, than the current drug; i.e.


A type I error would occur if we concluded that the two drugs produced different effects when in

fact there was no difference between them.



The following table gives a summary of possible results of any hypothesis test:

Decision

Reject H0 Don't reject H0

Truth

H0 Type I Error Right decision

H1 Right decision Type II Error

A type I error is often considered to be more serious, and therefore more important to avoid, than

a type II error. The hypothesis test procedure is therefore adjusted so that there is a guaranteed

'low' probability of rejecting the null hypothesis wrongly; this probability is never 0. This

probability of a type I error can be precisely computed as

P(type I error) = significance level =

The exact probability of a type II error is generally unknown.

If we do not reject the null hypothesis, it may still be false (a type II error) as the sample may not

be big enough to identify the falseness of the null hypothesis (especially if the truth is very close

to hypothesis).

For any given set of data, type I and type II errors are inversely related; the smaller the risk of

one, the higher the risk of the other.

A type I error can also be referred to as an error of the first kind.

Type II Error

In a hypothesis test, a type II error occurs when the null hypothesis H0, is not rejected when it is

in fact false. For example, in a clinical trial of a new drug, the null hypothesis might be that the

new drug is no better, on average, than the current drug; i.e.


A type II error would occur if it was concluded that the two drugs produced the same effect, i.e.

there is no difference between the two drugs on average, when in fact they produced different

ones.



A type II error is frequently due to sample sizes being too small.

The probability of a type II error is generally unknown, but is symbolized by and written

P(type II error) =

A type II error can also be referred to as an error of the second kind.

Test Statistic

A test statistic is a quantity calculated from our sample of data. Its value is used to decide

whether or not the null hypothesis should be rejected in our hypothesis test.

The choice of a test statistic will depend on the assumed probability model and the hypotheses

under question.

Critical Value(s)

The critical value(s) for a hypothesis test is a threshold to which the value of the test statistic in a

sample is compared to determine whether or not the null hypothesis is rejected.

The critical value for any hypothesis test depends on the significance level at which the test is

carried out, and whether the test is one-sided or two-sided.

Critical Region

The critical region CR, or rejection region RR, is a set of values of the test statistic for which the

null hypothesis is rejected in a hypothesis test. That is, the sample space for the test statistic is

partitioned into two regions; one region (the critical region) will lead us to reject the null

hypothesis H0, the other will not. So, if the observed value of the test statistic is a member of the

critical region, we conclude "Reject H0"; if it is not a member of the critical region then we

conclude "Do not reject H0".



Significance Level

The significance level of a statistical hypothesis test is a fixed probability of wrongly rejecting

the null hypothesis H0, if it is in fact true.

It is the probability of a type I error and is set by the investigator in relation to the consequences

of such an error. That is, we want to make the significance level as small as possible in order to

protect the null hypothesis and to prevent, as far as possible, the investigator from inadvertently

making false claims.

The significance level is usually denoted by

Significance Level = P(type I error) =

Usually, the significance level is chosen to be 0.05 (or equivalently, 5%).

P-Value

The probability value (p-value) of a statistical hypothesis test is the probability of getting a value

of the test statistic as extreme as or more extreme than that observed by chance alone, if the null

hypothesis H0, is true.

It is the probability of wrongly rejecting the null hypothesis if it is in fact true.

It is equal to the significance level of the test for which we would only just reject the null

hypothesis. The p-value is compared with the actual significance level of our test and, if it is

smaller, the result is significant. That is, if the null hypothesis were to be rejected at the 5%

significance level, this would be reported as "p < 0.05".

Small p-values suggest that the null hypothesis is unlikely to be true. The smaller it is, the more

convincing is the rejection of the null hypothesis. It indicates the strength of evidence for say,

rejecting the null hypothesis H0, rather than simply concluding "Reject H0' or "Do not reject

H0".

Power

The power of a statistical hypothesis test measures the test's ability to reject the null hypothesis

when it is actually false - that is, to make a correct decision.

In other words, the power of a hypothesis test is the probability of not committing a type II error.

It is calculated by subtracting the probability of a type II error from 1, usually expressed as:

Power = 1 - P(type II error) =



The maximum power a test can have is 1, the minimum is 0. Ideally we want a test to have high

power, close to 1.

One-sided Test

A one-sided test is a statistical hypothesis test in which the values for which we can reject the

null hypothesis, H0 are located entirely in one tail of the probability distribution.

In other words, the critical region for a one-sided test is the set of values less than the critical

value of the test, or the set of values greater than the critical value of the test.

A one-sided test is also referred to as a one-tailed test of significance.

The choice between a one-sided and a two-sided test is determined by the purpose of the

investigation or prior reasons for using a one-sided test.

Example

Suppose we wanted to test a manufacturer‘s claim that there are, on average, 50 matches in a

box. We could set up the following hypotheses

H0: µ = 50, against

H1: µ < 50 or H1: µ > 50

Either of these two alternative hypotheses would lead to a one-sided test. Presumably, we would

want to test the null hypothesis against the first alternative hypothesis since it would be useful to

know if there is likely to be less than 50 matches, on average, in a box (no one would complain if

they get the correct number of matches in a box or more).

Yet another alternative hypothesis could be tested against the same null, leading this time to a

two-sided test:


H1: µ not equal to 50

Here, nothing specific can be said about the average number of matches in a box; only that, if we

could reject the null hypothesis in our test, we would know that the average number of matches

in a box is likely to be less than or greater than 50.



Two-Sided Test

A two-sided test is a statistical hypothesis test in which the values for which we can reject the

null hypothesis, H0 are located in both tails of the probability distribution.

In other words, the critical region for a two-sided test is the set of values less than a first critical

value of the test and the set of values greater than a second critical value of the test.

A two-sided test is also referred to as a two-tailed test of significance.

The choice between a one-sided test and a two-sided test is determined by the purpose of the

investigation or prior reasons for using a one-sided test.

Example

Suppose we wanted to test a manufacturer‘s claim that there are, on average, 50 matches in a

box. We could set up the following hypotheses


H1: µ < 50 or H1: µ > 50

Either of these two alternative hypotheses would lead to a one-sided test. Presumably, we would

want to test the null hypothesis against the first alternative hypothesis since it would be useful to

know if there is likely to be less than 50 matches, on average, in a box (no one would complain if

they get the correct number of matches in a box or more).

Yet another alternative hypothesis could be tested against the same null, leading this time to a

two-sided test:


H1: µ not equal to 50

Here, nothing specific can be said about the average number of matches in a box; only that, if we

could reject the null hypothesis in our test, we would know that the average number of matches

in a box is likely to be less than or greater than 50.



T-TEST

The t-test assesses whether the means of two groups are statistically different from each other.

This analysis is appropriate whenever you want to compare the means of two groups.

Example: Sam hypothesizes that people who are allowed to sleep for only four hours will score

significantly lower than people who are allowed to sleep for eight hours on a cognitive skills test.

He brings sixteen participants into his sleep lab and randomly assigns them to one of two groups.

In one group he has participants sleep for eight hours and in the other group he has them sleep

for four. The next morning he administers the SCAT (Sam's Cognitive Ability Test) to all

participants. (Scores on the SCAT range from 1-9 with high scores representing better

performance).

SCAT scores

8 hours sleep

group (X) 5 7 5 3 5 3 3 9

4 hours sleep

group (Y) 8 1 4 6 6 4 1 2

X (x-Mx)2 y (y - My)

2

5 0 8 16

7 4 1 9

5 0 4 0

3 4 6 4

5 0 5 4

3 4 4 0

3 4 1 9

9 16 2 4

x=40 (x-

Mx)2=32

y=32 (y-

My)2=46

Mx=5

My=4



Interpretation: Sam's hypothesis was not confirmed. He did not find a significant difference

between those who slept for four hours versus those who slept for eight hours on cognitive test

performance.

One Sample t-test

A one sample t-test is a hypothesis test for answering questions about the mean where the data

are a random sample of independent observations from an underlying normal distribution N(µ,

), where is unknown.

The null hypothesis for the one sample t-test is:

H0: µ = µ0, where µ0 is known.

That is, the sample has been drawn from a population of given mean and unknown variance

(which therefore has to be estimated from the sample).

This null hypothesis, H0 is tested against one of the following alternative hypotheses, depending

on the question posed:

H1: µ is not equal to µ

H1: µ > µ

H1: µ < µ



Two Sample t-test

A two sample t-test is a hypothesis test for answering questions about the mean where the data

are collected from two random samples of independent observations, each from an underlying

normal distribution:

When carrying out a two sample t-test, it is usual to assume that the variances for the two

populations are equal, i.e.

The null hypothesis for the two sample t-test is:

H0: µ1 = µ2

That is, the two samples have both been drawn from the same population. This null hypothesis is

tested against one of the following alternative hypotheses, depending on the question posed.

H1: µ1 is not equal to µ2

H1: µ1 > µ2

H1: µ1 < µ2

Z Test

Description

The Z-test compares sample and population means to determine if there is a significant

difference.

It requires a simple random sample from a population with a Normal distribution and where

the mean is known.

Calculation

The z measure is calculated as:

z = (x - ) / SE

where x is the mean sample to be standardized, (mu) is the population mean and

SE is the standard error of the mean.

SE = / SQRT(n)



where is the population standard deviation and n is the sample size.

The z value is then looked up in a z-table. A negative z value means it is below the population

mean (the sign is ignored in the lookup table).

Discussion

The Z-test is typically with standardized tests, checking whether the scores from a particular

sample are within or outside the standard test performance.

The z value indicates the number of standard deviation units of the sample from the population

mean.

Comparing a Population Proportion to a Sample Proportion (Z-test)

Used to compare a proportion created by a random sample to a proportion originating from or

thought to represent the value for the entire population. As an example, to make sure your

random sample of 100 subjects is not biased regarding a person‘s sex, you could compare the

proportion of women in the sample to the known proportion of women in the underlying

population as reported in census data or by some other reliable source.

Example: Historical data indicates that about 10% of your agency's clients believe they were

given poor service. Now under new management for six months, a random sample of 110 clients

found that 15% believe they were given poor service.

Pu = .10

Ps = .15

n = 110

Assumptions

Independent random sampling

Nominal level data

Large sample size

State the Hypothesis



Ho: There is no statistically significant difference between the historical proportion of clients

reporting poor service and the current proportion of clients reporting poor service.

If 2-tailed test

Ha: There is a statistically significant difference between the historical proportion of clients

reporting poor service and the current proportion of clients reporting poor service.

If 1-tailed test

Ha: The proportion of current clients reporting poor service is significantly greater than the

historical proportion of clients reporting poor service.

Set the Rejection Criteria

Use z-distribution table to estimate critical value

If 2-tailed test, Alpha .05, Zcv = 1.96

If 1-tailed test, Alpha .05, Zcv = 1.65

Compute the Test Statistic

Estimate Standard Error

p = population proportion

q = 1 - p

n = sample size

Test Statistic



Decide Results of Null Hypothesis

If a 2-tailed test was used

Since the test statistic of 1.724 did not meet or exceed the critical value of 1.96, there is

insufficient evidence to conclude there is a statistically significant difference between the

historical proportion of clients reporting poor service and the current proportion of clients

reporting poor service.

If a 1-tailed test was used

Since the test statistic of 1.724 exceeds the critical value of 1.65, you can conclude the

proportion of current clients reporting poor service is significantly greater than the historical

proportion of clients reporting poor service.

Comparing Proportions From Two Independent Samples

(Z-test)

Used to compare two proportions created by two random samples or two subgroups of one

random sample.

EXAMPLE: A survey was conducted of students from the Princeton public school system to

determine if the incidence of hungry children was consistent in two schools located in lower-

income areas. A random sample of 80 elementary students from school A found that 23% did not

have breakfast before coming to school. A random sample of 180 elementary students from

school B found that 7% did not have breakfast before coming to school.

Assumptions

Independent random sampling

Nominal level data

Large sample size



State the Hypothesis

Ho: There is no statistically significant difference between the proportion of students in

school A not eating breakfast and the proportion of students in school B not eating

breakfast.

Ha: There is a statistically significant difference between the proportion of students in

school A not eating breakfast and the proportion of students in school B not eating

breakfast.

Set the Rejection Criteria

Use z-distribution table to estimate critical value

Alpha.05, Zcv = 1.96

Compute the Test Statistic

Estimate of Standard Error

where and

Test Statistic



Decide Results of the Null Hypothesis

Since the test statistic 3.721 exceeds the critical value of 1.96, you conclude there is a

statistically significant difference between the proportion of students in school A not

eating breakfast and the proportion of students in school B not eating breakfast.

The F-distribution is formed by the ratio of two independent chi-square variables divided by their

respective degrees of freedom.

Since F is formed by chi-square, many of the chi-square properties carry over to the F

distribution.

The F-values are all non-negative

The distribution is non-symmetric

The mean is approximately 1

There are two independent degrees of freedom, one for the numerator, and one for the

denominator.

There are many different F distributions, one for each pair of degrees of freedom.

F-Test

The F-test is designed to test if two population variances are equal. It does this by comparing the

ratio of two variances. So, if the variances are equal, the ratio of the variances will be 1.

If the null hypothesis is true, then the F test-statistic given above can be simplified

(dramatically). This ratio of sample variances will be test statistic used. If the null hypothesis is



false, then we will reject the null hypothesis that the ratio was equal to 1 and our assumption that

they were equal.

There are several different F-tables. Each one has a different level of significance. So, find the

correct level of significance first, and then look up the numerator degrees of freedom and the

denominator degrees of freedom to find the critical value.

You will notice that all of the tables only give level of significance for right tail tests. Because

the F distribution is not symmetric, and there are no negative values, you may not simply take

the opposite of the right critical value to find the left critical value. The way to find a left critical

value is to reverse the degrees of freedom, look up the right critical value, and then take the

reciprocal of this value. For example, the critical value with 0.05 on the left with 12 numerator

and 15 denominator degrees of freedom is found of taking the reciprocal of the critical value

with 0.05 on the right with 15 numerator and 12 denominator degrees of freedom.

Assumptions / Notes

The larger variance should always be placed in the numerator

The test statistic is F = s1^2 / s2^2 where s1^2 > s2^2

Divide alpha by 2 for a two tail test and then find the right critical value

If standard deviations are given instead of variances, they must be squared

When the degrees of freedom aren't given in the table, go with the value with the larger

critical value (this happens to be the smaller degrees of freedom). This is so that you are

less likely to reject in error (type I error)

The populations from which the samples were obtained must be normal.

The samples must be independent

EXAMPLE:

If we are not interested in whether one method is better compared to another, but were simply

trying to determine if the variances of were the same or different, we would need to use a 2-

tailed test. For instance, assume we made two sets of measurements of ethanol concentration in a

sample of vodka using the same instrument, but on two different days. On the first day, we found

a standard deviation of s1 = 9 ppm and on the next day we found s2 = 2 ppm. Both datasets

comprised 6 measurements. We want to know if we can combine the two datasets, or if there is a

significant difference between the datasets, and that we should discard one of them.

As usual, we begin by defining the null hypothesis, H0: σ12 = σ2

2, and the alternate hypothesis,

HA: σ12 ≠ σ2

2. The "≠" sign indicates that this is a 2-tailed test, because we are interested in both

cases: σ12 > σ2

2 and σ1

2 < σ2

2. For the F-test, you can perform a 2-tailed test by multiplying the



confidence level P by 2, so from a table for a 1-tailed test at the P = 0.05 confidence level, we

would perform a 2-tailed test at P = 0.10, or a 90% confidence level.

For this dataset, s2 > s1, Fcalc = s12/ s2

2 = 9

2/2

2 = 20.25. The tabulated value for ν = 5 at 90%

confidence is F5,5 = 5.050. Since Fcalc > F5,5, we reject the null hypothesis, and can say with 90%

certainty that there is a difference between the standard deviations of the two methods.

Tables for other confidence levels can be found in most statistics or analytical chemistry

textbooks. Be careful when using these tables, to pay attention to whether the table is for a 1- or

a 2-tailed test. In most cases, tables are given for 2-tailed tests, so you can divide by 2 for the 1-

tailed test. For the F-test, always ensure that the larger standard deviation is in the numerator, so

that F ≥ 1.

EXAMPLE 2:

In this example we test the equality of the variances of two data sets that belong to a normal

distribution. We start this example by creating 3 waves of different statistics. The first pair (data1

and data2) have the same variance but different means. The second pair (data2 and data3) have

the same mean but different variance. To create the data execute the commands:

Make/n=100 data1=100+gnoise(3)

Make/n=80 data2=80+gnoise(3)

Make/N=90 data3=80+gnoise(4)

Comparing the variance of two waves using a two-tailed hypothesis

To run the test execute the command:

StatsFTest/T=1/Q data1,data2

The results of the test appear in the F-Test table:

n1 100

Mean1 99.8754

Stdv1 3.39174

degreesOfFreedom1 99

n2 80

Mean2 79.6029

Stdv2 3.10709


F 1.19162

lowCriticalValue 0.659763

highCriticalValue 1.53104



P 0.418974

Accept 1

The F statistic is within the critical range so the two-tailed hypothesis of equal variances is

accepted.

Testing in the case of unequal variances (two tails test)

To run the test execute the following command:

StatsFTest/T=1/Q data1,data3

The results of the test appear in the F-Test table:

n1 100

Mean1 99.8754

Stdv1 3.39174


n2 80

Mean2 80.5489

Stdv2 4.43966


F 0.583641

lowCriticalValue 0.659763

highCriticalValue 1.53104

P 0.0112429

Accept 0

The rejection of H0 in this case is pretty sensitive to the choice of significance. It is apparent

from the P-value that it would have been accepted if alpha was set to 0.01.

One-tail testing for the same data

First H0: the variance of the first sample is greater than the variance of the second. To run the test

execute the command:

StatsFTest/T=1/Q/TAIL=1 data1,data3

n1 100

Mean1 99.8754

Stdv1 3.39174




n2 80

Mean2 80.5489

Stdv2 4.43966


F 0.583641

Critical 0.70553

P 0.00562143

Accept 0

H0 is rejected here as one would expect. Similarly,

StatsFTest/T=1/Q/TAIL=2 data1,data3

n1 100

Mean1 99.8754

Stdv1 3.39174


n2 80

Mean2 80.5489

Stdv2 4.43966


F 0.583641

Critical 1.4289

P 0.00562143

Accept 1

Here the F is smaller than the critical value so the two-tailed hypothesis can't be rejected.

Chi-Square Test

Chi-square is a statistical test commonly used to compare observed data with data we would

expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws,

you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8

males, then you might want to know about the "goodness to fit" between the observed and

expected. Were the deviations (differences between observed and expected) the result of chance,

or were they due to other factors. How much deviation can occur before you, the investigator,

must conclude that something other than chance is at work, causing the observed to differ from

the expected. The chi-square test is always testing what scientists call the null hypothesis, which

states that there is no significant difference between the expected and observed result.



The formula for calculating chi-square ( 2) is:

2= (o-e)

2/e

That is, chi-square is the sum of the squared difference between observed (o) and the expected

(e) data (or the deviation, d), divided by the expected data in all possible categories.

For example, suppose that a cross between two pea plants yields a population of 880 plants, 639

with green seeds and 241 with yellow seeds. You are asked to propose the genotypes of the

parents. Your hypothesis is that the allele for green is dominant to the allele for yellow and that

the parent plants were both heterozygous for this trait. If your hypothesis is true, then the

predicted ratio of offspring from this cross would be 3:1 (based on Mendel's laws) as predicted

from the results of the Punnett square (Figure B. 1).

Figure B.1 - Punnett Square. Predicted offspring from cross

between green and yellow-seeded plants. Green (G) is dominant

(3/4 green; 1/4 yellow).

To calculate 2 , first determine the number expected in each category. If the ratio is 3:1 and the

total number of observed individuals is 880, then the expected numerical values should be 660

green and 220 yellow.

Chi-square requires that you use numerical values, not percentages or ratios.

Then calculate 2 using this formula, as shown in Table B.1. Note that we get a value of 2.668

for 2. But what does this number mean? Here's how to interpret the

2 value:

1. Determine degrees of freedom (df). Degrees of freedom can be calculated as the number of

categories in the problem minus 1. In our example, there are two categories (green and yellow);

therefore, there is I degree of freedom.

2. Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis.

The relative standard commonly used in biological research is p > 0.05. The p value is the

probability that the deviation of the observed from that expected is due to chance alone (no other



forces acting). In this case, using p > 0.05, you would expect any deviation to be due to chance

alone 5% of the time or less.

3. Refer to a chi-square distribution table (Table B.2). Using the appropriate degrees of 'freedom,

locate the value closest to your calculated chi-square in the table. Determine the closestp

(probability) value associated with your chi-square and degrees of freedom. In this case (2=2.668), the p value is about 0.10, which means that there is a 10% probability that any

deviation from expected results is due to chance only. Based on our standard p > 0.05, this is

within the range of acceptable deviation. In terms of your hypothesis for this example, the

observed chi-squareis not significantly different from expected. The observed numbers are

consistent with those expected under Mendel's law.

Step-by-Step Procedure for Testing Your Hypothesis and Calculating Chi-Square

1. State the hypothesis being tested and the predicted results. Gather the data by conducting the

proper experiment (or, if working genetics problems, use the data provided in the problem).

2. Determine the expected numbers for each observational class. Remember to use numbers, not

percentages.

Chi-square should not be calculated if the expected value in any category is less than 5.

3. Calculate 2 using the formula. Complete all calculations to three significant digits. Round off

your answer to two significant digits.

4. Use the chi-square distribution table to determine significance of the value.

a. Determine degrees of freedom and locate the value in the appropriate column.

b. Locate the value closest to your calculated 2 on that degrees of freedom df row.

c. Move up the column to determine the p value.

5. State your conclusion in terms of your hypothesis.

a. If the p value for the calculated 2 is p > 0.05, accept your hypothesis. 'The deviation is

small enough that chance alone accounts for it. A p value of 0.6, for example, means that

there is a 60% probability that any deviation from expected is due to chance only. This is

within the range of acceptable deviation.

b. If the p value for the calculated 2 is p < 0.05, reject your hypothesis, and conclude that

some factor other than chance is operating for the deviation to be so great. For example, a

p value of 0.01 means that there is only a 1% chance that this deviation is due to chance

alone. Therefore, other factors must be involved.



The chi-square test will be used to test for the "goodness to fit" between observed and expected

data from several laboratory investigations in this lab manual.

Table 1

Calculating Chi-Square

Green Yellow

Observed (o) 639 241

Expected (e) 660 220

Deviation (o - e) -21 21

Deviation2 (d2) 441 441

d2/e 0.668 2

2 = d

2/e = 2.668 . .

Table 2 Chi-Square Distribution

Degrees

of

Freedom

(df)

Probability (p)

0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.01 0.001

1 0.004 0.02 0.06 0.15 0.46 1.07 1.64 2.71 3.84 6.64 10.83

2 0.10 0.21 0.45 0.71 1.39 2.41 3.22 4.60 5.99 9.21 13.82

3 0.35 0.58 1.01 1.42 2.37 3.66 4.64 6.25 7.82 11.34 16.27

4 0.71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.28 18.47

5 1.14 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09 20.52

6 1.63 2.20 3.07 3.83 5.35 7.23 8.56 10.64 12.59 16.81 22.46

7 2.17 2.83 3.82 4.67 6.35 8.38 9.80 12.02 14.07 18.48 24.32

8 2.73 3.49 4.59 5.53 7.34 9.52 11.03 13.36 15.51 20.09 26.12

9 3.32 4.17 5.38 6.39 8.34 10.66 12.24 14.68 16.92 21.67 27.88

10 3.94 4.86 6.18 7.27 9.34 11.78 13.44 15.99 18.31 23.21 29.59

Nonsignificant Significant



COMPARING POPULATION MEANS AND

PROPORTIONS

Sampling Distribution of the Differences between the Two Sample Means for Independent

Samples

When one wants to estimate the difference between two population means from independent

samples, then one will use a t-interval to calculate a confidence interval. If the sample variances

are not very different, one can use the pooled 2-sample t-interval.

Step 1. Find with df = n1 + n2 - 2.

Step 2. The endpoints of the (1 - ) 100% confidence interval for - is:

the degrees of freedom of t are n1 + n2 - 2.

Continuing from the previous example, give a 99% confidence interval for the difference

between the mean time it takes the new machine to pack ten cartons and the mean time it takes

the present machine to pack ten cartons.

Step 1. = 0.01, = t0.005 = 2.878, where the degrees of freedom are 18.

Step 2.

The 99% confidence interval is (-2.01, -0.17).

Interpret the above result:

We are 99% confident that - is between -2.01 and -0.17.



Paired samples:

The sample selected from the first population is related to the corresponding sample from the

second population.

When we use paired data we are looking at the difference of μ1 vs. μ2 - we write it μ1 - μ2 = μd. A

helpful chart for determining our hypothesis is:

Hypothesis Paired sample hypothesis p-value

H0: μ1 - μ2 = some hypothesized

value

H0: μd = some hypothesized

value

H0: μ1 - μ2 ≥ some hypothesized

value

Ha: μd > some hypothesized

value

area under the curve to the

right of the calculated t

H0: μ1 - μ2 ≤ some hypothesized

value

Ha: μd < some hypothesized

value

area under the curve to the

left of the calculated t

H0: μ1 - μ2 ≠ some hypothesized

value

Ha: μd ≠ some hypothesized

value

1) 2(area to the right oft) if t

is positive, or

2) 2( area to the left of t) if t

is negative

Trace metals in drinking water affect the flavor and an unusually high concentration can pose a

health hazard. Ten pairs of data were taken measuring zinc concentration in bottom water and

surface water. Does the data suggest that the true average concentration in the bottom water

exceeds that of surface water?

Location

1 2 3 4 5 6 7 8 9 10

Zinc

concentration

in

bottom water

.430 .266 .567 .531 .707 .716 .651 .589 .469 .723

Zinc

concentration

in

surface water

.415 .238 .390 .410 .605 .609 .632 .523 .411 .612



Paired t-Procedure:

Assumptions:

1. Paired samples

2. The differences of the pairs follow a normal distribution or the number of pairs is

large (note here that if the number of pairs is < 30, we need to check whether the

differences are normal, but we do not need to check for the normality of each

population)

Hypothesis:

Ho: - = 0

Ha: - 0

OR

Ho: - = 0

Ha: - < 0

OR

Ho: - = 0

Ha: - > 0

t-statistic:

Let = - ,

degrees of freedom = n - 1

where n denotes the number of pairs.

Paired t-interval:



An Example for the Paired t-Test

To perform a paired t-test for the previous trace metal example:

Assumptions:

1. Is this a paired sample?

Yes.

2. Is this a large sample?

No.

3. Since the sample size is not large enough (less than 30), we need to check whether the

differences follow a normal distribution.

F-test to compare Two Population Variances

In Lesson 10 I promised to show you how to statistically compare the variances of two

quantitative variables. We use the F-test to do this. When we test the equality of variances the

hypotheses of interest are:

Ho: 12 - 2

2 = 0

Ha: 12 - 2

2 0

An Example to Compare Two Population Variances

We return to the data on packaging time from Lesson 10. We are going to determine if it is

reasonable to assume that the two machines have equal population variances. Recall that the data

are given as:

New Machine Old Machine

42.1 41.3 42.4 43.2 41.8 42.7 43.8 42.5 43.1 44.0

41.0 41.8 42.8 42.3 42.7 43.6 43.3 43.5 41.7 44.1

= 42.14, s1 = 0.683 = 43.23, s2 = 0.750



Linear regression

Introduction to linear regression

Linear regression analyzes the relationship between two variables, X and Y. For each subject (or

experimental unit), you know both X and Y and you want to find the best straight line through

the data. In some situations, the slope and/or intercept have a scientific meaning. In other cases,

you use the linear regression line as a standard curve to find new values of X from Y, or Y from

X.

The term "regression", like many statistical terms, is used in statistics quite differently than it is

used in other contexts. The method was first used to examine the relationship between the

heights of fathers and sons. The two were related, of course, but the slope is less than 1.0. A tall

father tended to have sons shorter than himself; a short father tended to have sons taller than

himself. The height of sons regressed to the mean. The term "regression" is now used for many

sorts of curve fitting.

Prism determines and graphs the best-fit linear regression line, optionally including a 95%

confidence interval or 95% prediction interval bands. You may also force the line through a

particular point (usually the origin), calculate residuals, calculate a runs test, or compare the

slopes and intercepts of two or more regression lines.

In general, the goal of linear regression is to find the line that best predicts Y from X. Linear

regression does this by finding the line that minimizes the sum of the squares of the vertical

distances of the points from the line.

Note that linear regression does not test whether your data are linear (except via the runs test). It

assumes that your data are linear, and finds the slope and intercept that make a straight line best

fit your data.

Definitions

a) A scatterplot is a graph of paired X and Y values

b) A linear relationship is one in which the relationship between X and Y can best be

represented by a straight line.

c) A curvilinear relationship is one in which the relationship between X and Y can best be

represented by a curved line.

d) A perfect relationship exists when all of the points in the scatter plot fall exactly on the line

(or curve). An imperfect relationship is one in which there is a relationship, but not all

points fall on the line (or curve).



e) A positive relationship exists when Y increases as X increases (i.e., when the slope is

positive).

f) A negative relationship exists when Y decreases as X increases (i.e., when the slope is

negative).

r2, a measure of goodness-of-fit of linear regression

The value r2 is a fraction between 0.0 and 1.0, and has no units. An r2 value of 0.0 means that

knowing X does not help you predict Y. There is no linear relationship between X and Y, and the

best-fit line is a horizontal line going through the mean of all Y values. When r2 equals 1.0, all

points lie exactly on a straight line with no scatter. Knowing X lets you predict Y perfectly.

This figure demonstrates how Prism computes r2.

The left panel shows the best-fit linear regression line This lines minimizes the sum-of-squares

of the vertical distances of the points from the line. Those vertical distances are also shown on

the left panel of the figure. In this example, the sum of squares of those distances (SSreg) equals

0.86. Its units are the units of the Y-axis squared. To use this value as a measure of goodness-of-

fit, you must compare it to something.

The right half of the figure shows the null hypothesis -- a horizontal line through the mean of all

the Y values. Goodness-of-fit of this model (SStot) is also calculated as the sum of squares of the



vertical distances of the points from the line, 4.907 in this example. The ratio of the two sum-of-

squares values compares the regression model with the null hypothesis model. The equation to

compute r2 is shown in the figure. In this example r2 is 0.8248. The regression model fits the

data much better than the null hypothesis, so SSreg is much smaller than SStot, and r2 is near

1.0. If the regression model were not much better than the null hypothesis, r2 would be near zero.

You can think of r2 as the fraction of the total variance of Y that is "explained" by variation in X.

The value of r2 (unlike the regression line itself) would be the same if X and Y were swapped.

So r2 is also the fraction of the variance in X that is "explained" by variation in Y. In other

words, r2 is the fraction of the variation that is shared between X and Y.

In this example, 84% of the total variance in Y is "explained" by the linear regression model.

That leaves the rest of the vairance (16% of the total) as variability of the data from the model

(SStot)

The best fitting regression line: Least squares criterion

The formula for the least squares regression line is: Y Y Y ′ = b X + a

The most common version of the formula for the slope constant in least

squares regression is:

The ―Y‖ subscript on bY indicates that this is the slope for the regression of Y on X—that is for

the equation that allows prediction of Y-values from X-values.

SS is shorthand for the sum of the squared deviations about the

mean. The ―X‖ subscript indicates that it is the ―sum of squares‖ for the X-scores that we need.

The Regression Equation

[Standard notation: The data are pairs of independent and dependent variables {(xi,yi): i=1,...,n}.

The fitted equation is written is the predicted value of the response

obtained by using the equation. The residuals are the differences between the observed and the

predicted values . They are always calculated as (observed-predicted),

never the other way 'round.]



There are two primary reasons for fitting a regression equation to a set of data--first, to describe

the data; second, to predict the response from the carrier. The rationale behind the way the

regression line is calculated is best seen from the point-of-view of prediction. A line gives a good

fit to a set of data if the points are close to it. Where the points are not tightly grouped about any

line, a line gives a good fit if the points are closer to it than to any other line. For predictive

purposes, this means that the predicted values obtained by using the line should be close to the

values that were actually observed, that is, that the residuals should be small. Therefore, when

assessing the fit of a line, the vertical distances of the points to the line are the only distances that

matter. Perpendicular distances are not considered because errors are measured as vertical

distances, not perpendicular distances.

The simple linear regression equation is also called the least squares regression equation. Its

name tells us the criterion used to select the best fitting line, namely that the sum of the squares

of the residuals should be least. That is, the least squares regression equation is the line for which

the sum of squared residuals is a minimum.

It is not necessary to fit a large number of lines by trial-and-error to find the best fit. Some

algebra shows the sum of squared residuals will be minimized by the line for which

This can even be done by hand if need be.

When the analysis is performed by a statistical program package, the output will look something

like this.

A straight line can be fitted to any set of data. The formulas for the coefficients of the least

squares fit are the same for a sample, a population, or any arbitrary batch of numbers. However,



regression is usually used to let analysts generalize from the sample in hand to the population

from which the sample was drawn. There is a population regression equation,

0 + 1 X

and

Yi = 0 + 1 Xi + i,

where 0 and 1 are the population regression coefficients and i

is a random error peculiar to the i-th observation. Thus, each response is expressed as the sum of

a value predicted from the corresponding X, plus a random error.

The sample regression equation is an estimate of the population regression equation. Like any

other estimate, there is an uncertainty associated with it. The uncertainty is expressed in

confidence bands about the regression line. They have the same interpretation as the standard

error of the mean, except that the uncertainty varies according to the location along the line. The

uncertainty is least at the sample mean of the Xs and gets larger as the distance from the mean

increases. The regression line is like a stick nailed to a wall with some wiggle to it. The ends of

the stick will wiggle more than the center. The distance of the confidence bands from the

regression line is

,

where t is the appropriate percentile of the t distribution, se is the standard error of the estimate,

and x* is the location along the X-axis where the distance is being calculated. The distance is

smallest when x* = . These bands also

estimate the population mean value of Y

for X=x*.



There are also bands for predicting a single response at a particular value of X. The best estimate

is given, once again, by the regression line. The distance of the prediction bands from the

regression line is

.

For large samples, this is essentially tse, so the standard error of the estimate functions like a

standard deviation around the regression line.

The regression of X on Y is different

from the regression of Y on X. If

one wanted to predict lean body

mass from muscle strength, a new

model would have to be fitted

(dashed line). It could not be

obtained by taking the original

regression equation and solving for

strength. The reason is that in terms

of the original scatterplot, the best

equation for predicting lean body

mass minimizes the errors in the

horizontal direction rather than the

vertical. For example,

The regression of Strength

on LBM is

Strength = -13.971 + 3.016 LBM .

Solving for LBM gives

LBM = 4.632 + 0.332 Strength .

However, the regression of LBM on Strength is

LBM = 14.525 + 0.252 Strength .

The standard error of estimate:

There are in fact two such measures:

The variance error of estimate, and the standard error of estimate

The latter is the square root of the former. The standard error of estimate is to the least squares

regression line what the standard deviation is to the mean of a distribution.



The conceptual formula for the standard deviation of a distribution of Y-scores is given by:

The conceptual formula for the standard error of estimate when estimating Y given X is very

Similar.

Linear regression is used to predict a Y score from a score on X. Bear in mind the

following:

1) The relationship between X and Y must be linear. If the relationship is not linear,

prediction will not be very accurate.

2) Normally, we are not interested in predicting Y scores that are already known.

We derive our regression equation with sample data that consists of paired X and

Y scores, but use the equation to predict Y scores when only X values are given.

Because we use data collected from a sample to make these predictions, it is vital

to have a representative sample when deriving a regression equation.

3) A regression equation is properly used only for the range of the variables on

which it was based. We do not know whether the relationship between X and Y

continues to be linear beyond the range of sample values.

4) Prediction is most accurate if the data have the property of homoscedasticity—

i.e., if the variability of the Y scores is constant at all points along the regression

line.

5) When X and Y are both normally distributed and the number of paired scores is

large, the data in a bivariate frequency distribution often produce a so-called

bivariate normal distribution. When you have such a distribution, the standard

error of estimate can be used in the same way we used the standard deviation of

a normal distribution. That is, we could say that about 68% of the scores in the



scatterplot fall within 1 standard error of the regression line; and about 95% of

the scores fall within 2 standard errors of the regression line.

Linear regression and Pearson r

A correlation coefficient is a number that expresses the magnitude and direction of the

relationship between two (or more) variables. The Pearson product moment correlation

coefficient (Pearson r) is a correlation coefficient that ranges from - 1.00 to 1.00 .

a) if r = 1.00 -- perfect positive linear relationship

b) if r = -1.00 -- perfect negative linear relationship

c) if r = 0.00 -- no linear relationship

Pearson r from Raw Scores

Pearson r is a measure of the covariance of the X and Y scores, but is measured in

standard score units rather than in raw score units.

The coefficient of determination

If we square Pearson r, we obtain

r2 = proportion of the variability of Y accounted for by the linear relationship between

X and Y

Note that "the proportion of the variability of Y accounted for by the linear relationship between

X and Y" is usually abbreviated to "the variability of Y accounted for by X". Because r2

equals the proportion of the variability of Y accounted for by X, it is called the coefficient of

determination.

In order to understand what is meant by "proportion of variability of Y accounted for by X", let

us consider a concrete example where X is a score on a test of spelling competence, and Y is a



score on a test of writing ability. The data for N=6 subjects are shown below.

Let us imagine for just a moment that we do not know Mary‘s Y-score (writing ability), and wish

to predict it from her X-score. If there were no linear relationship between X and Y, then the

best prediction we could make would be the mean of the Y scores, or 50 in this case. However,

because there is a linear relationship between X and Y, we can do better than that by using the

regression line to make our prediction. As can be seen in Figure 4.1, Mary‘s predicted Y-score

is about 75.

Mary‘s actual Y-score is 90. Therefore, the deviation of her actual Y-score from the mean of Y

is 90 - 50 = 40.

Note that if you were to take the same kind of ( ) i Y − Y deviation score for each of the six

people,

square them, and add them up, you would have the sum of the squared deviations about the

mean, or SSY.

Returning to Mary‘s case, note that her deviation (from the mean of Y) score can be broken

down, or partitioned into two components. The first component is the deviation of her

predicted score from the mean of Y. As noted earlier, Mary‘s predicted score is 75, so:



The second component of the deviation of Mary‘s actual score from the mean of Y is the

deviation of her actual score from her predicted score:



CORRELATION

Definitions: Correlation is a statistical technique that is used to measure a relationship

between two variables. Correlation requires two scores from each individual (one score from

each of the two variables)

Distinguishing Characteristics of Correlation

Correlation procedures involve one sample containing all pairs of X and Y scores

Neither variable is called the independent or dependent variable

Use the individual pair of scores to create a scatterplot

Sum of Product Deviations

In the past you have used the sum of squares or SS to measure the amount of variation or

variability for a single variable The sum of products or SP provides a parallel procedure for

measuring the amount of covariation or covariability between two variables



Covariance

A measure of the linear association between variables

Positive indicates positive linear relationship

Negative indicates a negative linear relationship

Values close to zero indicates no linear relationship

It is dependent upon the units of measurement for x and y variables

Height in inches would give a larger covariance than height in feet; even with same

degree of association

So the magnitude of the covariance is not significant

Covariance Formula



Covariance Example

Correlation Coefficient Formula 1



Correlation Coefficient Formula 2



Using Dummy Variables in Regression

A dummy variable is a binary variable that has either 1 or zero. It is commonly used to examine

group and time effects in regression. Panel data analysis estimates the fixed effect and/or random

effect models using dummy variables. The fixed effect model examines difference in intercept

among groups, assuming the same slopes. By contrast, the random effect model estimates error

variances of groups, assuming the same intercept and slopes. An example of the random effect

model is the group wise heteroscesasticity model that assumes each group has different variances

One of the limitations of multiple-regression analysis is that it accommodates only

quantitative explanatory variables.

Dummy-variable regressors can be used to incorporate qualitative explanatory variables

into a linear model, substantially expanding the range of application of regression

analysis.

Introducing a Dummy Regressor

. One way of formulating the common-slope model is

where , called a dummy-variable regressor or an indicator variable, is

coded 1 for men and 0 for women:

Thus, for women the model becomes

and for men



How and Why Dummy Regression Works

Interpretation of parameters in the additive dummy-regression model: gives the difference in

intercepts for the two regression lines.Because these regression lines are parallel, also represents

the constant separation between the lines — the expected income advantage accruing to men

when education is held constant.

If men were disadvantaged relative to women, then would be negative.

gives the intercept for women, for whom = 0.

is the common within-gender education slope.

Figure 3 reveals the fundamental geometric ‗trick‘ underlying the coding of a dummy regressor:

We are, in fact, fitting a regression plane to the data, but the dummy regressor is defined only at

the values zero and one



Many Categories



EXAMPLE 2



Analysis of Variance:

An important technique for analyzing the effect of categorical factors on a response is to perform

an Analysis of Variance. An ANOVA decomposes the variability in the response variable

amongst the different factors. Depending upon the type of analysis, it may be important to

determine: (a) which factors have a significant effect on the response, and/or (b) how much of

the variability in the response variable is attributable to each factor

Assumptions:

For the validity of the F-test in ANOVA the following assumptions are made.

(i) The observations are independent

(ii) Parent population from which observations are taken is

normal and

(iii) Various treatment and environmental effects are additive in nature.



One way Classification:

Let us suppose that N observations xij , i = 1, 2, …… k ;j = 1,2….ni) of a random variable X are

grouped on some basis,into k classes of sizes n1, n2 , …..nk respectively

exhibited below

Test Procedure:

The steps involved in carrying out the analysis are:

1) Null Hypothesis:

The first step is to set up of a null hypothesis

H0: 1 = 2 = …= k

Alternative hypothesis H1: all i ‗ s are not equal (i = 1,2,…,k)

2) Level of significance : Let : 0.05



3) Test statistic:

Various sum of squares are obtained as follows.

a) Find the sum of values of all the (N) items of the given data. Let this grand total represented

by ‗ G‘ .Then correction factor

b) Find the sum of squares of all the individual items (xij)and then the Total sum of squares

(TSS) is

c) Find the sum of squares of all the class totals (or each treatment total) Ti (i:1,2,….k) and then

the sum of squares between the classes or between the treatments

(SST) is

Where ni (i: 1,2,…..k) is the number of observations in the ith class or number of observations

received by ith treatment

d) Find the sum of squares within the class or sum of squares due to error (SSE) by subtraction.

SSE = TSS - SST

4) Degrees of freedom (d.f):

The degrees of freedom for total sum of squares (TSS) is (N1). The degrees of freedom for SST

is (k1) and the degrees of freedom for SSE is (Nk)

5) Mean sum of squares:

The mean sum of squares for treatments is and meansum of squares for error is

6) ANOVA Table

The above sum of squares together with their respective degrees of freedom and mean sum of

squares will be summarized in the following table.



Calculation of variance ratio:

Variance ratio of F is the ratio between greater variance and smaller variance, thus

F = Variance between the treatments Variance within the treatment = MST/MSE

If variance within the treatment is more than the variance between the treatments, then numerator

and denominator should be interchanged and degrees of freedom adjusted accordingly.

7) Critical value of F or Table value of F:

The Critical value of F or table value of F is obtained from F table for (k-1, N-k) d.f at 5% level

of significance.

8) Inference:

If calculated F value is less than table value of F, we may accept our null hypothesis H0 and say

that there is no significant difference between treatments. If calculated F value is greater than

table value of F, we reject our H0 and say that the difference between treatments is significant.



Example 1:

SQUARES















EXAMPLE







Time series

Introduction: Arrangement of statistical data in chronological order i.e., in accordance with

occurrence of time, is known as ―Time Series‖. Such series have a unique important place in the

field of Economic and Business statistics. An economist is interested in estimating the likely

population in the coming year so that proper planning can be carried out with regard to food

supply, job for the people etc.

Similarly; a business man is interested in finding out his likely sales in the near future, so that the

businessman could adjust his production accordingly and avoid the possibility of inadequate

production to meet the demand. In this connection one usually deal with statistical data, which

are collected, observed or recorded at successive intervals of time. Such data are generally

referred to as‗time series‘.

Definition: A time series is a set of statistical observations arranged in chronological order.A

time series may be defined as a collection of readings belonging to different time periods, of

some economic variable or composite of variables. A time series is a set of observations of a

variable usually at equal intervals of time. Here time may be yearly, monthly, weekly, daily or

even hourly usually at equal intervals of time. Hourly temperature reading, daily sales, monthly

production are examples of time series

Components of Time series:

The components of a time series are the various elements which can be segregated from the

observed data. The following are the broad classification of these components.



In time series analysis, it is assumed that there is a multiplicative relationship between these four

components. Symbolically, Y = T S C I Where Y denotes the result of the four elements; T

= Trend ; S = Seasonal component; C = Cyclical components; I = Irregular component .In the

multiplicative model it is assumed that the four components are due to different causes but they

are not necessarily independent and they can affect one another.

Another approach is to treat each observation of a time

series as the sum of these four components. Symbolically

Y = T + S+ C+ I

The additive model assumes that all the components of the time series are independent of one

another.

1) Secular Trend or Long - Term movement or simply Trend

2) Seasonal Variation

3) Cyclical Variations

4) Irregular or erratic or random movements (fluctuations)

Secular Trend:

It is a long term movement in Time series. The general tendency of the time series is to increase

or decrease or stagnate during a long period of time is called the secular trend or simply trend.

Population growth, improved technological progress, changes in consumers taste are the various

factors of upward trend. We may notice downward trend relating to deaths, epidemics, due to

improved medical facilities and sanitations. Thus a time series shows fluctuations in the upward

or downward direction in the long run.

Methods of Measuring Trend:

Trend is measured by the following mathematical methods.

1. Graphical method

2. Method of Semi-averages

3. Method of moving averages

4. Method of Least Squares

Graphical Method:

This is the easiest and simplest method of measuring trend. In this method, given data must be

plotted on the graph, taking time on the horizontal axis and values on the vertical axis. Draw a

smooth curve which will show the direction of the trend. While fit ting a trend line the following

important points should be noted to get a perfect trend line.

The curve should be smooth.

As far as possible there must be equal number of points above and below the trend line.

The sum of the squares of the vertical deviations from the trend should be as small as

possible.



If there are cycles, equal number of cycles should be

above or below the trend line.

In case of cyclical data, the area of the cycles above and below should be nearly equal.



EXAMPLE





Merits:

1. It is simple and easy to calculate

2. By this method every one getting same trend line.

3. Since the line can be extended in both ways, we can find the later and earlier estimates.

Demerits:

1. This method assumes the presence of linear trend to the values of time series which may not

exist.

2. The trend values and the predicted values obtained by this method are not very reliable.



EXAMPLE

Even Period of Moving Averages:

When the moving period is even, the middle period of each set of values lies between the two

time points. So we must center the moving averages.

The steps are

Find the total for first 4 years and place it against the middle of the 2nd and 3rd year in

the third column.

Leave the first year value, and find the total of next four-year and place it between the 3rd

and 4th year.



Continue this process until the last value is taken.

Next, compute the total of the first two four year totals and place it against the 3rd year in

the fourth column.

Leave the first four years total and find the total of the next two four years‘ totals and

place it against the fourth year.

This process is continued till the last two four years‘ total is taken into account.

Divide this total by 8 (Since it is the total of 8 years) and put it in the fifth column.

These are the trend values.

EXAMPLE The production of Tea in India is given as follows. Calculate the Four-yearly moving averages



Merits:

The method is simple to understand and easy to adopt as compared to other methods.

It is very flexible in the sense that the addition of a few more figures to the data, the

entire calculations is not changed. We only get some more trend values.

Regular cyclical variations can be completely eliminated by a period of moving average

equal to the period of cycles

It is particularly effective if the trend of a series is very irregular.

Demerits:

It cannot be used for forecasting or predicting future trend,which is the main objective of

trend analysis

The choice of the period of moving average is sometimes subjective.

Moving averages are generally affected by extreme values of items.

It cannot eliminate irregular variations completely.

METHOD OF LEAST SQUARE:



EXAMPLE







Merits:

Since it is a mathematical method, it is not subjective so it eliminates personal bias of the

investigator.

By this method we can estimate the future values. As well as intermediate values of the

time series.

By this method we can find all the trend values.

Demerits:

It is a difficult method. Addition of new observations makes recalculations.

Assumption of straight line may sometimes be misleading since economics and business

time series are not linear.

It ignores cyclical, seasonal and irregular fluctuations.

The trend can estimate only for immediate future and not for distant future.

XBAR



Graphics Commands XBAR CHART DATAPLOT Reference Manual March 11, 1997 2-273

XBAR CHART

PURPOSE

Generates a mean control chart.

DESCRIPTION

An xbar (or mean) control chart is a data analysis technique for determining if a measurement

process has gone out of statistical control. The xbar chart is sensitive to shifts in location in the

measurement process.

It consists of:

Vertical axis = the mean for each sub-group.

Horizontal axis = sub-group designation.

In addition, horizontal lines are drawn at the overall mean and at the upper and lower control

limits. The distribution of the response variable is assumed to be normal. This assumption is the

basis for calculating the upper and lower control limits.

SYNTAX

XBAR CHART <y> <x> <SUBSET/EXCEPT/FOR qualification>

where <y> is the response (= dependent) variable (containing the raw data values);

<x> is an independent variable (containing the sub-group identifications);

and where the <SUBSET/EXCEPT/FOR qualification> is optional.

EXAMPLES

XBAR CHART Y X

XBAR CHART Y X SUBSET X > 2

NOTE

The attributes of the 4 traces can be controlled by the standard LINES, CHARACTERS, BARS,

and SPIKES commands. Trace 1 is the response variable, trace2 is the mean line, and traces 3

and 4 are the control limits. Some analysts prefer to draw the response variable as a spike or

character rather than a connected line.

DEFAULT

None

SYNONYMS

XBAR CONTROL CHART, MEAN CONTROL CHART, MEAN CHART, X CHART,

AVERAGE CONTROL CHART, and

AVERAGE CHART are synonyms for XBAR CHART.

RELATED COMMANDS

R CHART = Generates a range control chart.

S CHART = Generates a standard deviation control chart.

P CHART = Generates a p control chart.



NP CHART = Generates a Np control chart.

U CHART = Generates a U control chart.

C CHART = Generates a C control chart.

Q CONTROL CHART = Generates a Quesenberry style control chart.

CHARACTERS = Sets the types for plot characters.

LINES = Sets the types for plot lines.

SPIKES = Sets the on/off switches for plot spikes.

BARS = Sets the on/off switches for plot bars.

PLOT = Generates a data or function plot.

LAG PLOT = Generates a lag plot.

4-PLOT = Generates 4-plot univariate analysis.

MEAN PLOT = Generates a mean versus subset plot.

APPLICATIONS

Quality Control

IMPLEMENTATION DATE

88/2

XBAR CHART Graphics Commands

2-274 March 11, 1997 DATAPLOT Reference Manual

PROGRAM

SKIP 25

READ GEAR.DAT DIAMETER BATCH

.

LINE SOLID SOLID DOT DOT

TITLE AUTOMATIC

X1LABEL GROUP-ID

Y1LABEL MEAN

X CHART DIAMETER BATCH

1 2 3 4 5 6 7 8 9 10

0.99

0.995

1

1.005

1.01

XBAR CHART DIAMETER BATCH

GROUP-ID

MEAN XBAR





Documents

Business stats assignment