Part I

Quant 3610Quant 3610Weber State University

Dr. Stephen Hays RussellDr. Stephen Hays Russell

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Part I

Course Introduction

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Course IntroductionCourse Introduction

MeYouApproach in this courseMINITAB The Syllabus No Stat Text requiredDefinitions and refresher concepts

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DefinitionsDefinitions

Statistics (plural) A set of data

Statistics (singular) A science, a quantitative discipline concerned with the

methods of collection and analysis of data to facilitate wise decision making in the face of uncertainty.

Elementary units Persons or objects comprising a population of interest.

Population All elementary units of a defined group

Census An exhaustive survey on some characteristic of interest

of every elementary unit in a population

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Frame A complete listing of all elementary units

Parameter (usually Greek letters—like “ ” or “ ”) A number that describes a characteristic of the

population (“The Truth”) Sample

Any chosen subset of a population Simple Random Sample

A sample chosen in such a way that each elementary unit has an equal chance of being included

Statistic (usually Roman letters—like “x” or “b”) A numeric measure from sample information of some

characteristic of interest in a population

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Random Variable Any variable that takes on numeric values on the basis of chance

Inferential statistics Inferring the values of unknown population parameters from

sample statistics

Unbiased Estimator The expected value of the estimator equals the true value of the

parameter being estimated

Sampling distribution A listing of all possible values of a sample statistic along with the

associated probabilities of their occurrence. Sampling error

The difference between the sample statistic and the true parameter value

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Normal Probability Distribution Bell shaped, symmetrical, uniquely defined by

the mean and the standard deviation Z Table (Standard Normal Table) What proportion of normally distributed data lie

within + or – one standard deviation of the mean? Within + or – two standard deviations of the

mean?

Within + or – three standard deviations of the mean?

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Standard Normal Distribution Mean of Zero; Standard deviation of One

x

xxz

• Any normally distributed data can be converted to a standard normal

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Practice problemPractice problem

A manufacturer knows that the lifetime of its jet engine is normally distributed with a mean of 2,000 hours and a standard deviation of 100 hours. What is the probability that a randomly

chosen engine has a lifetime greater than 2,170 hours? Less than 1,840 hours? Between 2,000 and 2,075 hours?

Answers: .0446; .0548; .2734

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Another practice problem:Another practice problem:

A social biologist has established that the heights of adult men in the U.S. are normally distributed with a mean of 68 inches and a standard deviation of 4 inches. The tallest 10% of adult men are at least how

tall? (or put differently, 90% of all men are ____ inches tall or less.)

Answer: 73.1 inches (6’ 1”)

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Power Point Slides on the Web:Power Point Slides on the Web:

http://faculty.weber.edu/srussell

CLICK Quant 3610, Part I (etc.)

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MINITAB CommandsMINITAB Commands CDF

Finds the area under the normal curve between “minus infinity” and a specified Z

Example: CDF for –l.96 (= .025)

Example: CDF for 80; Normal 55, 6.

INVCDF Associates a specific area between “minus infinity” and an unknown Z Example: INVCDF for .025

(= -1.96) Example: INVCDF FOR .975

(= +1.96) Example: INVCDF for .9; Normal 68, 4.

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Two theorems in inferential statisticsTwo theorems in inferential statistics

Theorem One relates to populations that are normally distributed

Theorem Two—Called the Central Limit Theorem—relates to populations that are not normally distributed

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Theorem OneTheorem One

Given a normally distributed population:

If X (the mean of a simple random sample) is taken from a large population of X values and if the N population values are normally distributed, the sampling distribution of X is also normally distributed, regardless of sample size, n.

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Theorem TwoTheorem Two

Given a population that is not normally distributed, the Central Limit Theorem (CLT) states that:

If X (the mean of a simple random sample) is taken from a large population of X values and if the N population values are not normally distributed, the sampling distribution of X nevertheless approaches a normal distribution as sample size, n, increases.

Any sampling distribution of X is considered normal provided n 30 and also n < .05 N.

[Large sample assumption] [Large population assumption]

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Central Limit TheoremCentral Limit Theorem

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The Central Limit TheoremThe Central Limit Theorem

Perhaps the most important theorem in the entire field of statistical inference.

Consider what it says: As long as we take random samples that are

sufficiently large absolutely and fairly small relative to population size, we can consider the distribution of sample means to be a normal curve (and proceed to infer population parameters from sample statistics on that assumption).

We need not know or be concerned with the shape of the underlying population distribution (which is often unknowable given the problems with census taking).

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Sampling Distribution of the sample mean (X)Sampling Distribution of the sample mean (X)

The sampling distribution of the sample mean relates to the parent population like this:

X

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The sampling distribution of the sample mean relates to the parent population like this for the large population case:

nX

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Formula 1A Formula 1B

nX

When selections of sample elements are statistically independent events; typically referred to as "the large population case," because n < .05N

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Distribution of XDistribution of X

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The sampling distribution of the sample mean relates to the parent population like this for the small population case:

When selections of sample elements are statistically dependent events; typically referred to as "the small population case,” because n.05 N

Formula 1C Formula 1D

1

2

2

x N

nN

n

1

N

nN

nx

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Standard Normal Deviate for XStandard Normal Deviate for X

When X follows a normal distribution, X can be converted to a standard normal:

x

x

x

z

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Practice Problem with XPractice Problem with X

Mercury makes the Laser XRi engine used in speed boats. The company claims the engine delivers an average 220 horsepower and that the standard deviation of power delivered is 15 horsepower.

A potential boat manufacturer tests 100 randomly chosen XRi engines for single tests. If the claim is true, what is the probability that the sample mean will be less than 217 horsepower?

Answer: .0228

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Sampling distribution of the sample proportionSampling distribution of the sample proportion

The sample proportion is

The population proportion is .

nx

P nx

P

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Sampling distribution of the sample proportionSampling distribution of the sample proportion

The sample proportion follows a normal distribution per the CLT if large samples are taken from a large population:

If n5 and if n(1-) 5

we have a large sample.

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Sampling Distribution of the sample proportionSampling Distribution of the sample proportion

How the sampling distribution of the sample proportion (P) relates to the parent population:

P

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How the sampling distribution of the sample proportion (P) relates to the parent population large population case:

When selections of sample elements are statistically independent events; typically referred to as "the large population case,” because n < .05N

Formula 1E Formula 1F

nP

)1(2

nP

)1(

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How the sampling distribution of the sample proportion (P) relates to the parent population small population case:

When selections of sample elements are statistically dependent events; typically referred to as "the small population case,” because n .05 N

Formula 1G Formula 1H

1

)1(2N

nN

nP

1

)1(

N

nN

nP

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Standard Normal Deviate for PStandard Normal Deviate for P

When P follows a normal distribution, we can standardize this random variable:

p

p

P

Z

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Sample problem for PSample problem for P Honda claims that 20% of all prospective buyers of

minivans of any brand will be interested in their new 2005 Odyssey people mover.

You survey 200 randomly selected prospective buyers of minivans. You have them watch a Honda-produced promotional video on the new Odyssey and find that 33 of the prospective buyers express interest. Do you believe Honda’s claim based upon statistical analysis?

Answer: The probability of getting the sample result we obtained (or something more extreme) if their claim is true is .108

Z = (.165 - .2)/.0282843 = -1.2374

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Homework assignmentHomework assignment

Handout: Quant 3610 Minitab Practice Exercises Problem Set 1

Documents

Part I