© 2002 prentice-hall, inc.chap 7-1 business statistics: a first course 4th edition chapter 9...

© 2002 Prentice-Hall, Inc. Chap 7-1

Business Statistics: A First course

4th Edition

Chapter 9Fundamentals of Hypothesis Testing: One-Sample Tests

© 2002 Prentice-Hall, Inc. Chap 7-2

Chapter Topics

Hypothesis testing methodology Z test for the mean ( known) P-value approach to hypothesis testing Connection to confidence interval

estimation One-tail tests T test for the mean ( unknown) Z test for the proportion Potential hypothesis-testing pitfalls and

ethical considerations

© 2002 Prentice-Hall, Inc. Chap 7-3

What is a Hypothesis?

A hypothesis is a claim (assumption)about the populationparameter Examples of parameters

are population meanor proportion

The parameter mustbe identified beforeanalysis

I claim the mean GPA of this class is 3.5!

© 1984-1994 T/Maker Co.

© 2002 Prentice-Hall, Inc. Chap 7-4

The Null Hypothesis, H0

States the assumption (numerical) to be tested e.g.: The average number of TV sets in U.S.

Homes is at least three ( ) Is always about a population parameter

( ), not about a sample statistic ( )

0 : 3H

0 : 3H 0 : 3H X

© 2002 Prentice-Hall, Inc. Chap 7-5

The Null Hypothesis, H0

Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until

proven guilty Refers to the status quo Always contains the “=” sign May or may not be rejected

(continued)

© 2002 Prentice-Hall, Inc. Chap 7-6

The Alternative Hypothesis, H1

Is the opposite of the null hypothesis e.g.: The average number of TV sets in

U.S. homes is less than 3 ( ) Challenges the status quo Never contains the “=” sign May or may not be accepted Is generally the hypothesis that is

believed (or needed to be proven) to be true by the researcher

1 : 3H

© 2002 Prentice-Hall, Inc. Chap 7-7

Hypothesis Testing Process

Identify the Population

Assume thepopulation

mean age is 50.

( )

REJECT

Take a Sample

Null Hypothesis

No, not likely!

X 20 likely if Is ?

0 : 50H

20X

© 2002 Prentice-Hall, Inc. Chap 7-8

Sampling Distribution of

= 50

It is unlikely that we would get a sample mean of this value ...

... Therefore, we reject the

null hypothesis that μ = 50.

Reason for Rejecting H0

20

If H0 is trueX

... if in fact this were the population mean.

X

© 2002 Prentice-Hall, Inc. Chap 7-9

Filling Process

You are in charge of the filling process for the one pound jars of Cheese Whiz

You take a random sample of 36 jars each hour to determine if the filling process is in control

© 2002 Prentice-Hall, Inc.

Steps: State the Null Hypothesis (H0: )

State its opposite, the Alternative Hypothesis (H1: 16) Hypotheses are mutually exclusive &

exhaustive Sometimes it is easier to form the

alternative hypothesis first.

Identify the Problem

© 2002 Prentice-Hall, Inc. Chap 7-11

Decision Rule

Accept H0 if 15.8 oz X oz

Reject H0 if X < 15.8 oz or X > 16.2 oz • Assume a sample of 36 jars tested each hour

• Population standard deviation is .6 oz.

© 2002 Prentice-Hall, Inc. Chap 7-12

Possible Results

States of Nature

Correct Decision

Type II Error(

Correct Decision

Type I Error(

Do Not Reject Ho

Reject H0

ACTIONS

H0 is True = 16

H0 is False

© 2002 Prentice-Hall, Inc. Chap 7-13

Type I Error

X

Reject H0 Reject H0

Do Not Reject H0

16.21615.8

- 2 + 2 Z

.0228.0228.4772 .4772

Z

n

X

36

15.8

16 Z

36

16.2

16

© 2002 Prentice-Hall, Inc. Chap 7-14

Z1

36

15.8

15.9

Z2

36

16.2

15.9

15.8 15.9 16.2 X

Reject H0 Reject H0

Do Not Reject H0

.3413.4987

0 Z

.3413

.4987

.8400 =

© 2002 Prentice-Hall, Inc.

Reduce probability of one error and the other one goes up.

& Have an Inverse Relationship

© 2002 Prentice-Hall, Inc.

True Value of Population Parameter Increases When Difference Between Hypothesized Parameter

& True Value Decreases

Significance Level Increases When Decreases

Population Standard Deviation Increases When Increases

Sample Size n Increases When n Decreases

Factors Affecting Type II Error,

n

© 2002 Prentice-Hall, Inc. Chap 7-17

Hypothesis Testing: Steps

1. State H0 and H1

2. Choose - level of significance

3. Use Information in Steps 1 and 2 to Design a Decision Rule

4. Collect Data

5. Compare Sample Data to Decision Rule and State Conclusion.

© 2002 Prentice-Hall, Inc. Chap 7-18

Setting Up H0 and H1

H0 must contain the equality (=) Either H0 or H1 must precisely represent

what is being tested H0 and H1 must be mutually exhaustive

(must cover all possible outcomes)

© 2002 Prentice-Hall, Inc. Chap 7-19

Level of Significance,

Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of the sampling

distribution Is designated by , (level of

significance) Typical values are .01, .05, .10

Is selected by the researcher at the beginning

Provides the critical value(s) of the test

© 2002 Prentice-Hall, Inc. Chap 7-20

Level of Significance and the Rejection Region

H0: 3

H1: < 30

0

0

H0: 3

H1: > 3

H0: 3

H1: 3

/2

Critical Value(s)

Rejection Regions

© 2002 Prentice-Hall, Inc. Chap 7-21

Example: One Tail Test

Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.

368 gm.

H0: 368 H1: > 368

X

© 2002 Prentice-Hall, Inc. Chap 7-22

Finding Critical Value: One Tail

Z .04 .06

1.6 .9495 .9505 .9515

1.7 .9591 .9599 .9608

1.8 .9671 .9678 .9686

.9738 .9750

Z0 1.645

.05

1.9 .9744

Standardized Cumulative Normal Distribution Table

(Portion)

What is Z given = 0.05?

= .05

Critical Value = 1.645

.95

1Z

© 2002 Prentice-Hall, Inc. Chap 7-23

Example Solution: One Tail Test

= 0.5

n = 25

Critical Value: 1.645

Decision:

Conclusion:

Do Not Reject at = .05

No evidence that true mean is more than 368

Z0 1.645

.05

Reject

H0: 368 H1: > 368 1.50

XZ

n

1.50

© 2002 Prentice-Hall, Inc. Chap 7-24

p -Value Solution

Z0 1.50

P-Value =.0668

Z Value of Sample Statistic

From Z Table: Lookup 1.50 to Obtain .9332

Use the alternative hypothesis to find the direction of the rejection region.

1.0000 - .9332 .0668

p-Value is P(Z 1.50) = 0.0668

© 2002 Prentice-Hall, Inc. Chap 7-25

p -Value Solution(continued)

01.50

Z

Reject

(p-Value = 0.0668) ( = 0.05) Do Not Reject.

p Value = 0.0668

= 0.05

Test Statistic 1.50 is in the Do Not Reject Region

1.645

© 2002 Prentice-Hall, Inc. Chap 7-26

Example: Two-Tail Test

Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.

368 gm.

H0: 368

H1: 368

X

© 2002 Prentice-Hall, Inc. Chap 7-27

372.5 3681.50

1525

XZ

n

= 0.05

n = 25

Critical Value: ±1.96

Example Solution: Two-Tail Test

Test Statistic:

Decision:

Conclusion:

Do Not Reject at = .05

No Evidence that True Mean is Not 368Z0 1.96

.025

Reject

-1.96

.025

H0: 368

H1: 368

1.50

© 2002 Prentice-Hall, Inc. Chap 7-28

p-Value Solution

(p Value = 0.1336) ( = 0.05) Do Not Reject.

01.50

Z

Reject

= 0.05

1.96

p Value = 2 x 0.0668

Test Statistic 1.50 is in the Do Not Reject Region

Reject

© 2002 Prentice-Hall, Inc. Chap 7-29

For 372.5, 15 and 25,

the 95% confidence interval is:

372.5 1.96 15 / 25 372.5 1.96 15 / 25

or

366.62 378.38

If this interval contains the hypothesized mean (368),

we do not reject the null hypothesis.

I

X n

t does. Do not reject.

Connection to Confidence Intervals

© 2002 Prentice-Hall, Inc. Chap 7-30

t Test: Unknown

Assumption Population is normally distributed If not normal, requires a large sample

T test statistic with n-1 degrees of freedom

/

XtS n

© 2002 Prentice-Hall, Inc. Chap 7-31

Example: One-Tail t Test

Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, ands 15. Test at the 0.01 level.

368 gm.

H0: 368 H1: 368

is not given

© 2002 Prentice-Hall, Inc. Chap 7-32

Example Solution: One-Tail

= 0.01

n = 36, df = 35

Critical Value: 2.4377

Test Statistic:

Decision:

Conclusion:

Do Not Reject at = .01

No evidence that true mean is more than 368t35

0 2.4377

.01

Reject

H0: 368 H1: 368

372.5 3681.80

1536

Xt

Sn

1.80

© 2002 Prentice-Hall, Inc. Chap 7-33

p -Value Solution

01.80

t35

Reject

(p Value is between .025 and .05) ( = 0.01). Do Not Reject.

p Value = [.025, .05]

= 0.01

Test Statistic 1.80 is in the Do Not Reject Region

2.4377

© 2002 Prentice-Hall, Inc. Chap 7-34

PHStat | one-sample tests | t test for the mean, sigma known …

Example in excel spreadsheet

t Test: Unknown in PHStat

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 7-35

Involves categorical variables Fraction or % of population in a category If two categorical outcomes, binomial distribution

Either possesses or doesn’t possess the characteristic Sample proportion ( p )

Proportions

sizesample

successesofnumber

n

Xp

© 2002 Prentice-Hall, Inc. Chap 7-36

Example: Z Test for Proportion

• Problem: AstraZeneca claims that less than 5% of patients taking Nexium experience an upset stomach. • Approach: To test this claim, a random sample of 500 patients were interviewed. 15 of the patients experienced stomach pain.

• Solution: Test at the = .05 significance level.

© 2002 Prentice-Hall, Inc. Chap 7-37

= .05 n = 500 p = 15/500 = .03

Z Test for Proportion: Solution

H0: .05

H1: .05

Critical Value:

Reject at Reject at = .05Decision:

Conclusion:We do have sufficient evidence to support the claim that less than

5% of patients experience an upset stomach.

Z0

Reject

.05

Test Statistic:

Z p - (1 - )

n

=.03 -.05

.05 (1 - .05)500

= -2.05

© 2002 Prentice-Hall, Inc. Chap 7-38

= .05 n = 500 p = 15/500 = .03

Z Test for Proportion: Solution

H0: .05

H1: .05

Critical Value:

Reject at Reject at = .05Decision:

Conclusion:We do have sufficient evidence to support the claim that less than

5% of patients experience an upset stomach.

Z0

RejectHo

.05

Test Statistic:

Z p - (1 - )

n

=.03 -.05

.05 (1 - .05)500

= -2.05

© 2002 Prentice-Hall, Inc. Chap 7-39

Z Test for Proportion in PHStat

PHStat | one-sample tests | z test for the proportion …

Example in excel spreadsheet

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 7-40

Controlling and

A bank wishes to open a new branch if average monthly income is at least 4,000. It does not want to open the branch if average monthly income is less than or equal to 3,800.

Assume that

If the bank wants:What sample size should be used?What should the Decision Rule be?

© 2002 Prentice-Hall, Inc. Chap 7-41

H0: 4,00 H1: ≤ 3,800

n = 99Do not Reject H0 if Xc $Reject H0 if Xc < $3,917

2.33 =Xc - 3,800

500 n-1.645 =

Xc - 4,000

500 n

3,800

4,000 XZ

XZ

Xc

-1.645

Xc

2.330

0

Do not reject Ho

Do not reject H0

Reject H0

reject H0

.05 =

.01 =

© 2002 Prentice-Hall, Inc. Chap 7-42

Potential Pitfalls and Ethical Considerations

Randomize data collection method to reduce selection biases

Do not manipulate the treatment of human subjects without informed consent

Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance

© 2002 Prentice-Hall, Inc. Chap 7-43

Potential Pitfalls and Ethical Considerations

Do not practice “data cleansing” to hide observations that do not support a stated hypothesis

Report all pertinent findings

(continued)

© 2002 Prentice-Hall, Inc. Chap 7-44

Chapter Summary

Addressed hypothesis testing methodology

Performed Z Test for the mean ( Known)

Discussed p –Value approach to hypothesis

testing

Made connection to confidence interval

estimation

© 2002 Prentice-Hall, Inc. Chap 7-45

Chapter Summary

Performed one-tail and two-tail tests

Performed t test for the mean

( unknown)

Performed Z test for the proportion

Discussed potential pitfalls and ethical

considerations

(continued)