ECON 504 Chapter 0 - cba.edu.k · 2013. 3. 11. · ECON 504, CH 2 by M. Zainal 3 Sample Variance and Standard Deviation Let be a random sample of size and assume that the population

ECON 504, CH 2 by M. Zainal 1

DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS

A Review of Basic Statistical Concepts ECON 504

Chapter 0

Dr. Mohammad Zainal Spring 2013

A summary of basic statistical concepts that are related to

the subject of business statistics will be given in this chapter.

More specifically, the following concepts will be introduced

in brief.

It is the Definition of statistics

Descriptive summary of Numerical data

Graphical display of Numerical data

Probability Distributions

Binomial Probability Distribution

Normal Probability Distribution

Sampling Distributions

Statistical Inference based on one sample

Correlation Coefficient

Fitting a Straight Line

Assessing Normality

A Review of Basic Statistical Concepts


Definition of Statistics

General Meaning: Statistics means tables of vital data such as

number of births, deaths, marriages, divorces, car accidents, number

of telephone calls, electric shocks, etc.

Scientific Meaning: Statistics is the mathematics of collecting data,

tabulating, analyzing, interpreting these results, and then making

generalization to the population from the sample under study.

What is a Population?

A population is a collection of units that have the same

characteristics, which we are going to investigate.

What is a Sample?

A sample is a portion or a subset of the population.

Introduction to Forecasting

Descriptive summary of Numerical data.

Sample Mean

Let be a random sample of size. Then the Sample Mean is given by

It is unique for every numerical data set.

It takes information from all observations of the data set.

It is a useful measure to compare two or more data sets.

The sum of the deviations of the data from its mean is equal to zero. i.e.


xX

n

1

0n

i

i

x X


Sample Variance and Standard Deviation

Let be a random sample of size and assume that the population

mean is not known. Then the Sample Variance, is given by:

Note that sample standard deviation is equal

Coefficient of Variation

The coefficient of variation (CV) is defined to be the percentage of

the ratio of the standard deviation to the arithmetic mean, and is

given by:

It shows how large is the standard deviation compared to the mean.

It is very useful when comparing two different samples with

different means and standard deviations.


2 2

1

1( )

1

n

i

i

s x Xn

2s s

100s

CVX

Graphical display of Numerical data

Histogram

It is a graph in which the classes are marked on the horizontal axis and the frequencies on the vertical axis.

The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

Box Plot

It is a graphical display that describes the shape of a data set based on

quartiles.

To construct the box plot we need only five quantities:

The Minimum Value.

The First Quartile.

The Median.

The Third Quartile.

The Maximum Value.



It is also used to show the distance between the first and the third quartile

which is usually known as an interquartile range denoted by

Also, it is used to test for the existence of inconsistent values with the rest of

values in the data set which is usually known as outliers. Thus the outlier

should satisfy the following:

Dot plot

It is a graph in which the classes are marked on the horizontal axis and the observations are represented by dots placed above the axis.

Dot plots show the details of the data and also allow the forecaster to compare two or more sets of data.


3 1IQR Q Q

1 1.5Outlier Q IQR

3 1.5Outlier Q IQR

Scatter Diagram

It is a graph in which the relationships between two variables are

visualized. It is an X-Y plot.

Time Series Plot

It is a graph in which the data are plotted over time.

It reveals the variability of the data and the time at which peaks and

valleys occur and shows the relative size of the peaks and valleys

compares with rest of the series.

Remarks

Data collected on different elements at the same point in time then it said to

be a cross-section data. But if it is collected on the same element for the same

variable at different points in time or then it said to be a time-series data.

One important step in selecting an appropriate forecasting technique is to

identify the data patterns that that exist within a time series.

The correlogram or the autocorrelation function is a tool that usually used to

identify the pattern of the time series.



Probability Distributions

The researcher always starts inferring his possible results by conducting a statistical experiment concerning a phenomenon.

In fact the researcher is conducting a random experiment.

Experiment

An experiment is a process that leads to the occurrence of one result or outcome of several possible.

Random Experiment

A random experiment is a process that leads to the occurrence of different possible results or outcomes even if it is repeated different times under the same circumstances.

Random Variable

A random variable (RV) is a variable whose value is determined by the outcome of a random experiment.


Examples of RV

The quantity of monthly sales in a supper market, number of winning

games for a specific team in the world cup, number of monthly success

operations in heart center at a hospital, lifetime of a car batteries, the

value of the monthly telephone bills, number of a newborn male babies

in five cases entered a hospital …etc.

Note that these kinds of examples give their results by chance.

Therefore, the probability theory plays an important part in describing

these random variables, consequently the possible outcomes of the

random variable are represented by real numbers and described its

appearance by a probability value, or what is called the probability

distribution.

Consider the random experiment of tossing a fair coin two times, we

define the sample space as follows:


, , ,HH HT TH TT


Thus if we assume variable to represent the number of heads appeared. Then

Therefore, is called the random variable. Thus, mathematically we define the random variables as follows:

A random variable X is a real valued function defined on the sample

space Ω . i.e.

Examples on Random Variables:

The random variable X represents the number of a newborn male

babies in five cases entered a hospital.

The random variable X represents number of monthly success operations in heart center at a hospital.


0, 1, 2X

:X

0,1,2,3,4,5X

0,1,2,3,4,...X

The random variable X represents the quantity of monthly sales in

dollars in a supper market.

The random variable X represents the maximum temperature of any day

in a year in degrees Celsius.

Remarks:

Note that the random variables in examples (1) and (2) are defined on a discrete interval which show a separated values, and this type of random variables known as Discrete Random Variables.

Also, note that the random variables in examples (3) and (4) are defined on a continuous interval which shows non separated values, and this type of random variables known as Continuous Random Variables.


:0 X x x

: X x x


Probability Distribution

A probability distribution (X, P(X)) is a real valued function that assigns to each random variable a probability value between 0 and 1. i.e.

where is [0,1] a subset of R, the set of the real numbers.

Depending on the nature of the random variables, we can define two types of probability distributions:

Discrete Probability Distribution.

Continuous Probability Distribution.

Discrete Probability Distribution

For the RV , the probability of X = xi for i = 1… n

denoted by P(xi) = P(X= xi) for i = 1… n is called a discrete probability

distribution for the RV X if the following two conditions are satisfied:


: [0,1]P X

1 2 , , , nX x x x

Note that P(xi) = P(X= xi) for i = 1… n is called the probability function of X.

The quantity P(X ≤ c) is called the cumulative distribution function of and is defined by

Also the probability distribution of the random variable X can be written in the form of tables as follows:


( ) 0; 1,2, ,iP x i n

1

( ) 1n

i

i

P x

( ) ( )i

i

x c

P X c P x

1 2

1 2( ) ( ) ( ) ( )

n

n

X x x x

P X P x P x P x



Continuous Probability Distribution

For the random variable , the probability of function X , denoted by , is called a continuous probability distribution for the random variable X if the following two conditions are satisfied:

1. 2.

Note that is called the probability function of X .

Remark:

The quantity is called the cumulative distribution function X of and is defined by

For simplicity we call it the distribution function.

( ) 0;f x x ( ) 1f x dx

( );f x x

:X x x

( );f x x

( ) ( )F c P X c

( ) ( ) ( )

c

F c P X c f x dx


Mathematical Expectations and Variance:

In this section, we will study the mathematical expectation and

the variance of the random variable for discrete probability

distributions.

For the following discrete probability distribution of a random

variable X:

We can define the mean, variance and standard deviation as

follows:

The mathematical expectation for the random variable X , denoted by

. is defined by

Note that is also known as the mean of the random variable X

1 2

1 2( ) ( ) ( ) ( )

n

n

X x x x

P X P x P x P x

( )E X ( ) ( )E X XP X

)(XE



The variance of the random variable X , denoted by , is

defined by

The standard deviation of the random variable X , denoted by

is defined by

2 ( )Var X 2 2( ) ( ) ( )Var X X P X

( )SD X

( ) ( )SD X Var X


A Binomial probability distribution is a probability of getting r successes in n independent trials in which the sample is drawn with replacement from an infinite population and the probability of success remains the same for each trial.

For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula

where n = total number of trials, p = probability of success, q = probability of failure 1 – p, x = number of successes in n trials, n – x = number of failures in n trials

The cumulative distribution function (CDF) of a binomial random variable X is given by

( )

!; 0,1,2, ,

!( )!

r n r

n r

r n r

P r C p q

np q r n

r n r

0

!( )

!( )!

cr n r

n

nP X c p q

r n r



A Normal Probability Distribution is an important continuous distribution in statistics in which most of the data sets related to economics, commerce, management, physical and social sciences follow its behavior.

It is the most widely used distribution in statistical analysis.

This distribution is characterized by two parameters the mean and the variance .

Definition:

Assume that X is a random variable follows a normal probability distribution with mean and variance . Then the probability function of this distribution can be written as follows:

where and

2

2

2

1( ) ;

2

x

f X e X

0


The normal probability distribution, when plotted, gives a bell-shaped curve such that

The total area under the curve is 1.0.

The curve is symmetric around the mean.

The two tails of the curve extended indefinitely.



Finding Areas Under the Normal Curve :

As we mentioned earlier, the normal probability distribution can be expressed by a mathematical function

in which the probability that x falls between a and b is the integral of the above function from a to b, i.e.

21

21( )

2

xb

a

P a x b e dx

21

21( )

2

x

f x e


Using the substitution

We obtain

Note that Z is normally distributed with mean 0 and variance 1,

which is usually known as a standard normal distribution.

But as you know, we will not use the formula to find the probability. instead, we will use the standard normal table.

XZ

21( )

2

b

z

a

a bP a X b e dz P Z



Empirical Rule:


Sampling Distributions

In this subject we will concentrate in defining the sampling distributions and the use of central limit theorem for sample mean.

Sampling Distribution of Sample Mean

It is the probability distribution of the population of all possible sample means obtained from samples of size n .

It is easily shown that for infinite or large population, the sample mean has mean and variance .

The question now is what is the sampling distribution of the sample mean ?

The Central Limit Theorem gives the answer.

X 2 n

X



Central Limit Theorem

For large samples (generally n > 30) from any population, the sample mean will follow an approximate normal distribution. i.e.

Or equivalently,

Remark

When obtaining small samples (generally n ≤ 30) from a normal population, the sample mean may not be a normal distribution and the population standard deviation is estimated by the sample standard deviation.

Then the ratio is distributed as a Student's t distribution

with n-1 degrees of freedom.

X

(0,1)X

Z Normal

n

2

,X Normaln

X

Xt

s n


Statistical Inference based on one sample

Statistical Inference:

A statistical inference is the science of using a sample of measurements to make generalizations about a population.

It actually consists of estimation and testing hypothesis.

The estimation is also consists of two types point estimations and confidence intervals.

We will concentrate in confidence intervals.

Confidence Interval (CI):

A Confidence Interval for a population parameter is an interval constructed around the point estimate so that we confident, that the population parameter is inside the interval.

By computing such an interval, we estimate how far the point estimate might be from the parameter.



Confidence Intervals for the mean of a normal population when is known

A 100 (1-) % CI for the population mean is given by

The Length of the Confidence Interval is

The Sample size n is estimated by

where B is the known as the maximum errors or the error bound.

2 2a aX Z X Zn n

22 aL Zn

2

2aZn

B


Hypothesis Testing:

A hypothesis testing is a statistical procedure used to provide evidence in favor of some statement (called a hypothesis).

Hypothesis testing might be used to assess whether a population parameter, such as a population parameter (i.e. mean, median, proportion, variance … etc.) differs from a specified standard or previous values.

Types of Hypotheses

The Null Hypothesis Ho, which is a statement concerning a population parameter, the researcher wishes to discredit this statement.

The Alternative Hypothesis Ha, which is a statement in contradiction to the null hypothesis, the researcher wishes to support this statement.

Types of Errors

Type I error, which occurs if you rejected Ho when in fact it is true.

Type II error, which occurs if you accepted Ho when in fact it is untrue.



Definition 1

The level of significance is the probability of rejecting Ho when Ho is true. i.e.

= P (Type I error)

= P (Rejecting Ho| Ho is true)

Definition 2

The power of the test is the probability of rejecting Ho when Ho is false. i.e.

Power = 1-

Where

= P (Type II error)

= P (Accepting Ho | Ho is false)


Two hypothesis

Consider an example of a person who has been indicted for committing a crime and is being tried in a court. Based on the available evidence, the judge will make one of the following decisions:

The person is not guilty (null hypothesis, Ho).

The person is guilty (alternative hypothesis, Ha).

A null hypothesis is a claim (or statement) about a population parameter that is assumed to be true until it is declared false.

An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.

The alternative hypothesis is often what the test is attempting to establish.

The research hypothesis should be expressed as the alternative hypothesis.



The conclusion that the research hypothesis is true comes from sample data

that contradict the null hypothesis.

Manufacturers’ claims are usually given the benefit of the doubt and stated

as the null hypothesis.

The conclusion that the claim is false comes from sample data that

contradict the null hypothesis.

In the court example, we start the prosecution by assuming the person is

not guilty.

The prosecutors collect all possible evidences to prove the null hypothesis

is false (guilty).

31

ECON 504, CH 2 by M. Zainal


Rejection and nonrejection regions

Not enough evidence to declare the

person guilty and, hence, the null

hypothesis is not rejected in this region

Enough evidence to declare the person

guilty and, hence, the null hypothesis is

rejected in this region

Nonrejection region Rejection region

C

Critical point

Level of

evidence 0

Actual Situation

The person is not guilty

The person is guilty

Court’s decision

The person is not guilty

Correct decision Type II or

error

The person is guilty Type I or

error Correct decision



A contingency table illustrates the possible outcomes of a

statistical hypothesis test.

Correct

Decision

Type II Error or

error

Correct

Decision Type I Error or

error Reject H0

Accept H0

H0 True H0 False

Conclusion

Actual Situation


Definition 3

A test statistic is a rule which leads to a decision to accept or to reject the hypothesis under consideration when the experimental sample values have been obtained.

Types of Test Statistics

The Z-Test Statistic

we use this test if the following conditions are satisfied

The population is normal.

Large sample size n for both cases of known or unknown. (Note that we replace by s for the unknown case).

Small sample size n when is known.

0XZ

n



The t-Test Statistic

we use this test if the following conditions are satisfied

The population is normal.

Small sample size n when is unknown. (Note that we replace by s

for this case).

0Xt

s n


Now running these tests, we reject Ho if:

Another way of accepting or rejecting the null hypothesis is

based on the concept of probability value or what is called the

p-Value.

Number Type of Test Rejection Region

Z-Test t-Test

1

Ho: o

Ha: o

2

Ho: ≤ o

Ha: > o

3

Ho: ≥ o

Ha: < o

2aZ Z

Z Z

Z Z

2, 1nt t

, 1nt t

, 1nt t



Definition

The p-Value is the actual probability of getting the sample mean value if the null hypothesis is true.

The Decision

Reject Ho if p

Do not reject Ho if p >

Calculating the p-Value

p = P (Z-test | Ho is true)

p = P (t-Test | Ho is true)


Correlation

Correlation is a statistical method used to determine whether a relationship between variables exists.

Simple Linear Correlation (Pearson Correlation Coefficient)

where

xy

xx yy

SR

S S

1 1

1 1

2 2

1 12 2 2 2

1 1 1 1

( )( )

( ) , ( )

n n

i in ni i

xy i i i i

i i

n n

i in n n ni i

xx i i yy i i

i i i i

x y

S x x y y x yn

x y

S x x x S y y yn n



Properties

* -1 ≤ R ≤ 1 * R 1 or R -1 (+ve or –ve strong relationship)

* R 0 (No Relationship)

Testing For Population Correlation Coefficient

Using level of significant , we test for the significance of the

correlation coefficient. i.e.

Test Statistic

Decision

Reject Ho if: or

0 : 0

: 0a

H

H

2

2

1

nt R

R

2, 2nt t p value


Fitting a Straight Line

For each scatter plot, we draw different lines to represent data. But we should choose the line of the best fit.

Thus the method of least squares gives us the best fitting straight line.

The observed Model

where

is the - intercept ( the mean value of when =0, and

is the slope ( the amount increase in or decrease in the

mean value of y associated with a one-unit increase in x.

is the error term.

0i i i iy x

0

i

i



Model Assumptions

The errors 's must be:

Statistically Independent

Normally Distributed

Zero Mean

Constant Variance

i


The Best Fitting Line

Where

,

and

0i i iy b b x

1

xy

xx

Sb

S 0 1b y b x

1 1

1 1

2

12 2

1 1

2

12 2

1 1

( )( )

( )

( )

n n

i in ni i

xy i i i i

i i

n

in ni

xx i i

i i

n

in ni

yy i i

i i

x y

S x x y y x yn

x

S x x xn

y

S y y yn



Assessing Normality

Different methods are offered to check whether a set of data follow a normal distribution or not.

Some of them are as follows:

One may look at the Histogram

One may look at Anderson-Darling normality test (See Minitab)

One may look at the Sample autocorrelation function (ACF)

One may look at the goodness of fit test for the data set

One may look at Kolmogorov-Smirnov test for normality.

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ECON 504 Chapter 0 - cba.edu.k · 2013. 3. 11. · ECON 504, CH 2 by M. Zainal 3 Sample Variance and Standard Deviation Let be a random sample of size and assume that the population