Chapter 9 Student Lecture Notes 9-1 · Chapter 9 Student Lecture Notes 9-1 © 2004 Prentice-Hall,...

Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.

Chapter 9 Student Lecture Notes 9-1

© 2004 Prentice-Hall, Inc. Chap 9-1

Basic Business Statistics(9th Edition)

Chapter 9Fundamentals of Hypothesis Testing: One-Sample Tests


Chapter TopicsHypothesis Testing MethodologyZ Test for the Mean ( Known)p-Value Approach to Hypothesis TestingConnection to Confidence Interval EstimationOne-Tail Testst Test for the Mean ( Unknown)Z Test for the Proportion

Test for the Variance or Standard DeviationPotential Hypothesis-Testing Pitfalls and Ethical Issues

σ

σ

2χ


What is a Hypothesis?

A Hypothesis is a Claim (Assertion)about the PopulationParameter

Examples of parametersare population mean (µ)or proportion (P )The parameter mustbe identified beforeanalysis

I claim the mean GPA of this class is 3.5!

© 1984-1994 T/Maker Co.

µ =




The Null Hypothesis, H0

States the Claim or Assertion to be TestedE.g., The mean GPA is 3.5

Null Hypothesis is Always about a Population Parameter ( ), Not about a Sample Statistic ( ) Is the Hypothesis a Researcher Tries to Reject

0 : 3.5H µ =

0 : 3.5H µ =0 : 3.5H X =


The Null Hypothesis, H0

Begin with the Assumption that the Null Hypothesis is True

Similar to the notion of innocent untilproven guilty

Refers to the Status QuoAlways Contains the “=” SignThe Null Hypothesis May or May Not be Rejected

(continued)


The Alternative Hypothesis, H1

Is the Opposite of the Null HypothesisE.g., The mean GPA is NOT 3.5 ( )

Challenges the Status QuoNever Contains the “=” SignIs Generally the Hypothesis that the Researcher is Interested in

1 : 3.5H µ ≠




Error in Making Decisions

Type I Error Reject a true null hypothesis

When the null hypothesis is rejected, we can say that “We have shown the null hypothesis to be false (with some ‘slight’ probability, i.e.

, of making a wrong decision)

Has serious consequencesProbability of Type I Error is

Called level of significanceSet by researcher

α

α


Error in Making Decisions

Type II ErrorFail to reject a false null hypothesisProbability of making a Type II Error is

The Probability of Not Making a Type II Error

Called the Power of the Test

Probability of Not Making Type I Error

Called the Confidence Coefficient( )1 α−

(continued)

( )1 β−

β


Hypothesis Testing Process

Identify the Population

Assume thepopulation

mean GPA is 3.5( )

REJECT

Take a Sample

Null Hypothesis

No, not likely!X 2.4 likely if Is 3.5?µ= =

0 : 3.5H µ =

( )2.4X =




= 3.5

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

... if in fact this werethe population mean.

... Therefore, we reject the

null hypothesis that = 3.5.

Reason for Rejecting H0

µ

Sampling Distribution of

2.4If H0 is true

X

X

µ


Level of Significance,

Defines Unlikely Values of Sample Statistic if Null Hypothesis is True

Called rejection region of the sampling distribution

Designated by , (level of significance)Typical values are .01, .05, .10

Selected by the Researcher at the BeginningControls the Probability of Committing a Type I ErrorProvides the Critical Value(s) of the Test

α

α


Level of Significance and the Rejection Region

H0: µ ≥ 3.5 H1: µ < 3.5

0

0

0

H0: µ ≤ 3.5 H1: µ > 3.5

H0: µ = 3.5 H1: µ ≠ 3.5

α

α

α/2

Critical Value(s)

Rejection Regions




Result ProbabilitiesH0: Innocent

The Truth The Truth

Verdict Innocent Guilty Decision H0True H0 False

Innocent Correct Decision

Type II Error

Do NotReject

H0

Type IIError

Guilty Type I Error

CorrectDecision

RejectH0

Type IError

Power

Jury Trial Hypothesis Test

( )1 α−Confidence

( )1 β−

( )β

( )α


Type I & II Errors Have an Inverse Relationship

α

β

Reduce probability of one error and the other one goes up holding everything else unchanged.


Factors Affecting Type II Error

True Value of Population Parameterincreases when the difference between the

hypothesized parameter and its true value decreases

Significance Levelincreases when decreases

Population Standard Deviationincreases when increases

Sample Sizeincreases when n decreases

β

β

α

β σ

β n

α

β

β

β σ




How to Choose between Type I and Type II Errors

Choice Depends on the Cost of the ErrorsChoose Smaller Type I Error When the Cost of Rejecting the Maintained Hypothesis is High

A criminal trial: convicting an innocent personThe Exxon Valdez: causing an oil tanker to sink

Choose Larger Type I Error When You Have an Interest in Changing the Status Quo

A decision in a startup company about a new piece of softwareA decision about unequal pay for a covered group


Critical Values Approach to Testing

Convert Sample Statistic (e.g., ) to Test Statistic (e.g., Z, or t statistic)Obtain Critical Value(s) for a Specifiedfrom a Table or Computer

If the test statistic falls in the critical region, reject H0

Otherwise, do not reject H0

X

α


p-Value Approach to TestingConvert Sample Statistic (e.g., ) to Test Statistic (e.g., Z, or t statistic)Obtain the p-value from a table or computer

p-value: probability of obtaining a test statistic as extreme or more extreme ( or ) than the observed sample value given H0 is trueCalled observed level of significanceSmallest value of that an H0 can be rejected

Compare the p-value withIf p-value , do not reject H0

If p-value , reject H0

X

≤ ≥

<≥ α

α

αα




General Steps in Hypothesis Testing

E.g., Test the Assumption that the True Mean # of TV Sets in U.S. Homes is at Least 3 ( Known)σ

1. State the H0

2. State the H1

3. Choose

4. Choose n

5. Choose Test

0

1

: 3: 3

=.05100

Z

HH

ntest

µµ

α

≥<

=α


100 households surveyed

Computed Z =-2,

p-value = .0228

Reject null hypothesis

The true mean # TV set is less than 3

(continued)

Reject H0

α

-1.645 Z

6. Set up critical value(s)

7. Collect data

8. Compute test statistic and p-value

9. Make statistical decision

10.Express conclusion

General Steps in Hypothesis Testing


One-Tail Z Test for Mean( Known)

AssumptionsPopulation is normally distributedIf not normal, requires large samplesNull hypothesis has or sign only

is known

Z Test Statistic

σ

≤ ≥

/XZ

nµ

σ−

=

σ




Rejection Region

Z0

Reject H0

Z0

Reject H0

H0: µ ≥ µ0H1: µ < µ0

H0: µ ≤ µ0H1: µ > µ0

Z must be significantlybelow 0 to reject H0

Small values of Z don’t contradict H0 ; don’t

reject H0 !

αα


Example: One-Tail Test

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 σ to be 15 grams. Test at the α = 0.05 level.

368 gm.

H0: µ ≤ 368 H1: µ > 368

X


Reject and Do Not Reject Regions

Z0 1.645

.05

Reject

1.50

X368Xµ = 372.5

0 : 368H µ ≤

1 : 368H µ >

Do Not Reject




Finding Critical Value: One-Tail

Z .04 .06

1.6 .9495 .9505 .9515

1.7 .9591 .9599 .9608

1.8 .9671 .9678 .9686

.9738 .9750Z0 1.645

.05

1.9 .9744

Standardized Cumulative Normal Distribution Table

(Portion)What is Z given α = 0.05?

α = .05

Critical Value = 1.645

.95

1Zσ =


Example Solution: One-Tail Test

α = 0.5n = 25Critical Value: 1.645

Test Statistic:

Decision:

Conclusion:Do Not Reject at α = .05.

Insufficient Evidence that True Mean is More Than 368.

Z0 1.645

.05Reject

H0: µ ≤ 368 H1: µ > 368

1.50XZ

n

µσ

−= =

1.50


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




p -Value Solution(continued)

01.50

Z

Reject

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

p Value = 0.0668

α = 0.05

Test Statistic 1.50 is in the Do Not Reject Region

1.645


One-Tail Z Test for Mean( Known) in PHStat

PHStat | One-Sample Tests | Z Test for the Mean, Sigma Known …Example in Excel Spreadsheet

σ


Example: Two-Tail Test

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

368 gm.

H0: µ = 368 H1: µ ≠ 368

X





Z0 1.96

.025

Reject

-1.96

.025

1.50

X368Xµ = 372.5

Reject

0 : 368H µ =

1 : 368H µ ≠


372.5 368 1.501525

XZ

n

µσ

− −= = =α = 0.05

n = 25Critical Value: ±1.96

Example Solution: Two-Tail Test

Test Statistic:

Decision:

Conclusion:Do Not Reject at α = .05.

Z0 1.96

.025

Reject

-1.96

.025

H0: µ = 368 H1: µ ≠ 368

1.50

Insufficient Evidence that True Mean is Not 368.


p-Value Solution


0 1.50 Z

Reject

α = 0.05

1.96

p-Value = 2 x 0.0668


Reject




PHStat | One-Sample Tests | Z Test for the Mean, Sigma Known …Example in Excel Spreadsheet

Two-Tail Z Test for Mean( Known) in PHStatσ


( ) ( )

For 372.5, 15 and 25,the 95% confidence interval is:

372.5 1.96 15/ 25 372.5 1.96 15/ 25or

366.62 378.38

X nσ

µ

µ

= = =

− ≤ ≤ +

≤ ≤

Connection to Confidence Intervals

We are 95% confident that the population mean is between 366.62 and 378.38.

If this interval contains the hypothesized mean (368),we do not reject the null hypothesis.

It does. Do not reject.


t Test: Unknown

AssumptionPopulation is normally distributedIf not normal, requires a large sample

is unknown

t Test Statistic with n-1 Degrees of Freedom

σ

/XtS n

µ−=

σ




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, and s = 15. Test at the α = 0.01 level.

368 gm.

H0: µ ≤ 368 H1: µ > 368σ is not given



t350 2.4377

.01

Reject

1.80

X368Xµ = 372.5

0 : 368H µ ≤

1 : 368H µ >

Do Not Reject


Example Solution: One-Tail

α = 0.01n = 36, df = 35Critical Value: 2.4377

Test Statistic:

Decision:

Conclusion:Do Not Reject at a = .01.

t350 2.4377

.01

Reject

H0: µ ≤ 368 H1: µ > 368

372.5 368 1.801536

Xt Sn

µ− −= = =

1.80

Insufficient Evidence that True Mean is More Than 368.




p -Value Solution

0 1.80t35

Reject

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

p-Value = [.025, .05]

α = 0.01

Test Statistic 1.80 is in the Do Not Reject Region2.4377


PHStat | One-Sample Tests | t Test for the Mean, Sigma Known …Example in Excel Spreadsheet

t Test: Unknown in PHStatσ


Proportion

Involves Categorical VariablesTwo Possible Outcomes

“Success” (possesses a certain characteristic) and “Failure” (does not possess a certain characteristic)

Fraction or Proportion of Population in the “Success” Category is Denoted by p




Proportion

Sample Proportion in the Success Category is Denoted by pS

When Both np and n(1-p) are at Least 5, pSCan Be Approximated by a Normal Distribution with Mean and Standard Deviation

(continued)

Number of SuccessesSample Sizes

Xpn

= =

sp pµ =(1 )

spp p

nσ −

=


Example: Z Test for Proportion

( )

( ) ( )

Check:500 .04 20

51 500 1 .04

480 5

np

n p

= =

≥

− = −

= ≥

A marketing company claims that a survey will have a 4% response rate. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the α= .05 significance level.



Z0 1.96

.025

Reject

-1.96

.025

1.1411

0.04SP pµ = = 0.05

Reject

0 : 0.04H p =

1 : 0.04H p ≠

SP




0.05

Critical Values: ± 1.96

1.1411

( ) ( ).05 .04 1.1411

1 .04 1 .04500

Sp pZp p

n

− −≅ = =

− −

Z Test for Proportion: Solution

α = .05n = 500

Do not reject at α = .05.

H0: p = .04 H1: p ≠ .04

Test Statistic:

Decision:

Conclusion:

Z0

Reject Reject

.025.025

1.96-1.96

We do not have sufficient evidence to reject the company’s claim of 4% response rate.

SP0.04


p -Value Solution


0 1.1411 Z

Reject

α = 0.05

1.96

p-Value = 2 x .1269


Reject


Z Test for Proportion in PHStat

PHStat | One-Sample Tests | Z Test for the Proportion …Example in Excel Spreadsheet




Test for Variance or Standard Deviation

AssumptionPopulation is normally distributed

Test Statistic

2χ

( ) 22

2

2

2

1

where sample size sample variance hypothesized population variance

n S

nS

χσ

σ

−=

=

=

=


Example: Test for Standard Deviation

368 gm.

2χ

Has the standard deviation of the weight of cereal boxes produced by a production process changed from the specified level of 15 grams? A sample of 25 cereal boxes shows a sample standard deviation of 17.7 grams. Test at a 5% level of significance.

0

2

12

: 15 grams

( 225 grams squared): 15 grams

( 225 grams squared)

H

H

σ

σσ

σ

=

=≠

≠


Test for Standard Deviation: Solution

2χ

0

1

: 15 grams: 15 grams

HH

σσ

=≠

0.05 25nα = =Critical Values: 12.401 and 39.364

RejectReject

0.0250.025

39.36412.401

0.95

( ) ( ) ( )( )

222

22

1 25 1 17.715

33.42

n Sχ

σ− −

= =

=

Test Statistic:

33.42

Decision:Do not reject at 0.05α =

Conclusion:There is insufficient evidence that the standard deviation of the process has changed from the specified level of 15 grams




Test for Variance or Standard Deviation in PHStat

PHStat | One-Sample Test | Chi-Square Test for Variance …Example in Excel Spreadsheet

2χ

Microsoft Excel Worksheet


Potential Pitfalls andEthical Issues

Data Collection Method is Not Randomized to Reduce Selection Biases

Treatment of Human Subjects are Manipulated Without Informed Consent

Data Snooping is Used to Choose between One-Tail and Two-Tail Tests, and to Determine the Level of Significance


Potential Pitfalls andEthical Issues

Data Cleansing is Practiced to Hide Observations that do not Support a Stated Hypothesis

Fail to Report Pertinent Findings

(continued)




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

σ


Chapter Summary

Performed One-Tail and Two-Tail Tests

Performed t Test for the Mean ( Unknown)

Performed Z Test for the Proportion

Performed Test for Variance or Standard

Deviation

Discussed Potential Pitfalls and Ethical Issues

(continued)

σ

2χ

Chapter 9 Student Lecture Notes 9-1 · Chapter 9 Student Lecture Notes 9-1 © 2004 Prentice-Hall,...

Documents

Transcript of Chapter 9 Student Lecture Notes 9-1 · Chapter 9 Student Lecture Notes 9-1 © 2004 Prentice-Hall,...