Chap 08 Student

8/7/2019 Chap 08 Student

1/39

1999 Prentice-Hall, Inc. Chap. 8 - 1

Statistics for Managers

Using Microsoft Excel

Chapter 8

Fundamentals of HypothesisTesting: One-Sample Tests


2/39


Chapter Topics

Hypothesis Testing Methodology

Z Test for the Mean (WKnown)

p-Value Approach to Hypothesis Testing

Connection to Confidence Interval Estimation

One Tail Test

t Test of Hypothesis for the Mean

Z Test of Hypothesis for the Proportion


3/39


A hypothesis is an

assumption about the

population parameter.

A parameter is a

characteristic of the

population, like its

mean or variance.

The parameter must be

identified before

analysis.

I assume the mean GPA

of this class is 3.5!

1984-1994 T/Maker Co.

What is a Hypothesis?


4/39


States the Assumption (numerical) to be tested

e.g. The grade point average of juniors is at

least 3.0 (H0: Qu

3.0)

Begin with the assumption that the null

hypothesis is TRUE.

(Similar to the notion of innocent until proven guilty)

The Null Hypothesis, H0

Refers to the Status Quo

Always contains the = sign

The Null Hypothesis may or may not be rejected.


5/39


Is the opposite of the null hypothesise.g. The grade point average of juniors isless than 3.0 (H1: Q < 3.0)

Challenges the Status Quo

Never contains the = sign

The Alternative Hypothesis may or maynot be accepted

Is generally the hypothesis that isbelieved to be true by the researcher

TheAlternative Hypothesis, H1


6/39


Steps:

State the Null Hypothesis (H0: Q u3.0)

State its opposite, the Alternative

Hypothesis (H1: Q < 3.0)

Hypotheses are mutually exclusive &

exhaustiveSometimes it is easier to form the

alternative hypothesis first.

Identify the Problem


7/39


Population

Assume the

population

mean age is 50.

(Null Hypothesis)

REJECT

The Sample

Mean Is 20

SampleNull Hypothesis

50?20 !$! QXIs

Hypothesis Testing Process

No, not likely!


8/39


Sample MeanQ = 50

Sampling Distribution

It is unlikely

that we wouldget a sample

mean of this

value ...

... if in fact this were

the population mean.

... Therefore, wereject the null

hypothesis that

Q = 50.

20H0

Reason for Rejecting H0


9/39


Defines Unlikely Values of Sample

Statistic if Null Hypothesis Is True

Called Rejection Region of SamplingDistribution

Designated E (alpha)

Typical values are 0.01, 0.05, 0.10

Selected by the Researcher at the Start

Provides the Critical Value(s) of the Test

Level of Significance, E


10/39


Level of Significance, Eandthe Rejection Region

H0: Q u3

H1: Q < 3

0

0

0

H0: Q e 3

H1: Q > 3

H0: Q !3

H1: Q { 3

E

E

E/2

Critical

Value(s)

RejectionRegions


11/39


Type I Error

Reject True Null Hypothesis (False Positive)

Has Serious Consequences

Probability of Type I Error Is E

Called Level of Significance

Set by researcher

Type II Error

Do Not Reject False Null Hypothesis (False

Negative)

Probability of Type II Error Is F (Beta)

Errors in Making Decisions


12/39


H0: Innocent

Jury Trial Hypothesis Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0 False

Innocent Correct ErrorDo Not

Reject

H0

1 - EType II

Error (F )

Guilty Error CorrectReject

H0

Type IError(E )

Power

(1 - F)

Result Possibilities


13/39


E

F

Reduce probability ofone error

and the other one goes up.

E& F Have anInverse Relationship


14/39


True Value of Population Parameter

Increases When Difference Between Hypothesized

Parameter & True Value Decreases

Significance Level E Increases When EDecreases

Population Standard Deviation W Increases When W Increases

FactorsAffecting

Type II Error, F

E

F

F W


15/39


True Value of Population Parameter

Increases When Difference Between Hypothesized

Parameter & True Value Decreases

Significance Level E Increases When EDecreases

Population Standard Deviation W Increases When W Increases

Sample Size n Increases When n Decreases

FactorsAffecting

Type II Error, F

E

F

F W

F

n


16/39


Choice depends on the cost of the error

Choose little type I error when the cost of

rejecting the maintained hypothesis is high A criminal trial: convicting an innocent person

The Exxon Valdise: Causing an oil tanker to sink

Choose large type I error when you have

an interest in changing the status quo

A decision in a startup company about a new piece of

software

Adecision about unequal pay for a covered group.

How to choose between Type I and

Type II errors


17/39


Deregulation leads to lower air travel

prices. The university discriminates against

women faculty members

Stock prices increase when a firm

announces a layoff.

Example Hypotheses:

How do you test them?


18/39


Hiring Policy Hypotheses

A recruiter must decide who to hire. This

is like forming a hypothesis about

whether or not the individual predicts tobe a good employee. Assume that an

employee is either good or poor.

WHAT ARE THE NULL ANDALTERNATIVE HYPOTHESES?


19/39


20/39



WHAT ARE THE POSSIBLE ERRORS

THAT THE RECRUITER CAN MAKE?

HOW DO THESE RELATE TO TYPE I

AND TYPE II ERRORS?


21/39



FAILURE TO HIRE A GOOD EMPLOYEE

(failure to reject a false null=type II

error)

FAILURE TO REJECT A POOR

EMPLOYEE(rejecting a null when it is

really true is type I error)


22/39



A positive decision is a decision to reject

the null. A false positive is therefore a

type I error (hiring a poor person).

A negative decision is a failure to reject

the null. A false negative is therefore a

type II error (not hiring a good person)


23/39



The addition of more criteria should

increase your ability to distinguish poor

candidates => type I error falls

However, more criteria mean that more

good employees are cut accidently =>

type II error increases.When do you use more criteria?


24/39


Convert Sample Statistic (e.g., ) to test

statistic, for example a Z, t or F-statistic

Compare to Critical value obtained from

a table.

If the test statistic falls in the Critical

Region, RejectH0; OtherwiseDo Not Reject

H0

Critical Value of the test statistic

X


25/39


Probability of Obtaining a Test Statistic

More Extreme e or u) than Actual

Sample Value Given H0 Is True

Called Observed Level of Significance

Smallest Value of a H0 Can Be Rejected

Used to Make Rejection Decision

Ifp value uE Do Not Reject H0

Ifp value < E, Reject H0

p Value Test


26/39


1. StateH0 H0 : Qu3.0

2. StateH1 H1 : Q

3. Choose E E = .05

4. Choose n n = 100

5. Choose Test: t Test (or p Value)

Hypothesis Testing: Steps

Test theAssumption that the true mean

grade point average of juniors is at least 3.


27/39


6. Set Up Critical Value(s) t = -1.7

7. Collect Data 100 students sampled

8. Compute Test Statistic Computed Test Stat.= -2

(computed P value=.04, two-tailed test)

9. Make Statistical Decision Reject NullHypothesis

10. Express Decision The true mean grade point is lessthan 3.0

Hypothesis Testing: Steps

Test theAssumption that grade point average of

juniors is at least 3.

(continued)


28/39


Z0

E

Reject H0

Z0

Reject H0

E

H0: QuH1: Q < 0

H0: Qe0H1: Q > 0

Must BeSignificantly

BelowQ= 0Small values dont contradictH0

Dont Reject H0!

Rejection Region


29/39


Assumptions

Population is normally distributed

If not normal, only slightly skewed & a largesample taken (Central limit theorem applies)

Parametric test procedure

t test statistic, with n-1 degrees of freedom

t-Test: WUnknown

n

St

Q!


30/39


# in sample - number of parameters that must be

estimated before test statistic can be computed.

For a single sample t-test, we must first estimate the

mean before we can estimate the standard deviation.

Once the mean is estimated, n-1 of the values are left

since we know that the nth value is equal to

Degrees of Freedom

f

!

1

1

n

iixxn


31/39


32/39


PH Stat Entries


33/39


80.1

3615

3685.372!

!

!

n

S

Xt

Q

Example Solution: One Tail


34/39


Involves categorical variables

Fraction or % of population in a category

Iftwo categorical outcomes, binomial

distribution

Either possesses or doesnt possess the characteristic

Sample proportion (ps)

Proportions

sizesample

successesofnumber

nps !!


35/39


The null hypothesis for the proportion also implies we

know the variance, since the variance is just P times (1-

P).

This is a good approximation when the sample size is

large.

If the sample size is small, we could use the binomial

distribution to compute the exact p value that a sampleof size n would yield a sample proportionps given the

population proportion P. Using the normal

approximation is much easier.

Z test for proportions


36/39


Example:Z Test for Proportion

Problem:Amarketing company claims

that it receives 4% responses from its

Mailing.

Approach: To test this claim, a random

sample of 500 were surveyed with 25

responses.Solution: Test at the E = .05 significance

level.


37/39


PH STAT Entries


38/39


Z Test for Proportion:

Solution

Critical Values: s 1.96Decision:

Conclusion:We do not have sufficient

evidence to reject the companys

claim of 4% response rate.Z0

Reject Reject

.025.025

a

= 1.14


39/39


Chapter Summary

Addressed Hypothesis Testing Methodology

Performed t- Test for the Mean (Wunknown)

Discussed p-ValueApproach to Hypothesis Testing

Performed One Tail and Two Tail Tests

Performed Z Test of Hypothesis for the Proportion

Documents

Chap 08 Student