©2006 Thomson/South-Western 1 Chapter 10 – Hypothesis Testing for the Mean of a Population Slides...

©2006 Thomson/South-Western 1

Chapter 10 –Chapter 10 –

Hypothesis Hypothesis Testing for the Testing for the Mean of a Mean of a PopulationPopulation

Slides prepared by Jeff HeylLincoln University

©2006 Thomson/South-Western

Concise Managerial StatisticsConcise Managerial Statistics

KVANLIPAVURKEELING

KVANLIPAVURKEELING


Hypothesis Testing on the Hypothesis Testing on the MeanMean

Null hypothesis Null hypothesis ((HHoo)): A statement : A statement (equality or inequality) concerning a (equality or inequality) concerning a population parameter; the researcher population parameter; the researcher wishes to discredit this statement.wishes to discredit this statement.

Alternative hypothesis Alternative hypothesis ((HHaa)): A : A statement in contradiction to the null statement in contradiction to the null hypothesis; the researcher wishes to hypothesis; the researcher wishes to support this statement.support this statement.


Type I and Type II ErrorsType I and Type II Errors

= probability of rejecting the Ho when Ho is true (Type I error)

= probability of failing to rejecting the Ho when Ho is false (Type II error)

Actual Situation

Conclusion Ho True Ho False

Fail to Reject Ho Correct decision Type II errorType II error

Reject Ho Type I errorType I error Correct decision


Hypothesis Testing ProcessHypothesis Testing Process

Determine the HDetermine the Hoo and H and Haa

Determine the significance levelDetermine the significance level

Compare the sample mean (variance) Compare the sample mean (variance) to the hypothesized mean (variance)to the hypothesized mean (variance)

Decide whether to fail to reject or Decide whether to fail to reject or reject Hreject Hoo

Determine what the decision means in Determine what the decision means in reference to the problemreference to the problem


Height ExampleHeight Example

HHoo: : = 5.9 = 5.9

HHaa: : ≠≠ 5.9 5.9

= .05 == .05 = P(rejecting H P(rejecting Hoo when H when Hoo is true) is true) critical value critical value = ± 1.96= ± 1.96

RejectReject HHoo if if > 1.96 > 1.96XX - 5.9 - 5.9

/ / nnXX - 5.9 - 5.9

/ / nn



Figure 10.1

Z-k 0 k

|Z| > k

.025 .025

Area = .5 - .025 = .475



Figure 10.2

X5.79’ µx = 5.9’

distance = .11’ = 2.38 x .046

x ≈ = .046’.4 75


Computed ValueComputed Value

BecauseBecause - 2.38 < - 1.96, - 2.38 < - 1.96, we reject Hwe reject Hoo thus wethus we

conclude that the average population male conclude that the average population male height is not equal toheight is not equal to 5.9 5.9

ZZ = = = -2.38 = = = = -2.38 = ZZ **XX - 5.9 - 5.9

/ / nn5.79 - 5.95.79 - 5.9

.4 / 75.4 / 75



Figure 10.3

Z-k 0

Area = .01

k

Area = .5 - .005 = .495

.005



Figure 10.4

Z * = -2.38

Z-2.575 2.5750

Area = .01

Reject Ho if Z * falls here

Reject Ho if Z * falls here


Hypothesis TestingHypothesis Testing5 Step Procedure5 Step Procedure

3.3. Define the rejection region Define the rejection region 4.4. Calculate the test statisticCalculate the test statistic5.5. Give a conclusion in terms of the problemGive a conclusion in terms of the problem

1.1. Set up the null and alternative hypothesisSet up the null and alternative hypothesis

ZZ = =XX - µ - µoo

/ / nn

2.2. Define the test statisticDefine the test statistic


Everglo Light ExampleEverglo Light Example

1.1. Define the hypotheses Define the hypotheses

HHoo: µ = 400: µ = 400 H Haa: µ ≠ 400: µ ≠ 400

3.3. Define the rejection region Define the rejection region

reject Hreject Hoo if Z if Z > 1.645> 1.645 or Z or Z < -1.645< -1.645


ZZ = = XX - 400 - 400

/ / nn



Figure 10.5

Z-1.645 1.6450

Area = = .1

Area = .5 - .05 = .45

2

= .05 2

= .05

Z * = 2.5



5.5. State the conclusion State the conclusion

There is sufficient evidence to conclude There is sufficient evidence to conclude that the average lifetime of Everglo bulbs that the average lifetime of Everglo bulbs is not 400 hoursis not 400 hours

4.4. Calculate the value of the test statisticCalculate the value of the test statistic

ZZ ** ≈ = = 2.5 ≈ = = 2.5202088

420 - 400420 - 40040 / 2540 / 25


Confidence Intervals and Confidence Intervals and Hypothesis TestingHypothesis Testing

When testing HWhen testing Hoo: µ = µ: µ = µoo versus H versus Haa: µ ≠ µ: µ ≠ µoo using the five-step procedure and a using the five-step procedure and a significance level, significance level, , H, Hoo will be rejected if will be rejected if and only if µand only if µoo lies outside the lies outside the (1 - (1 - ) • 100) • 100 confidence interval for confidence interval for µµ

X X -- k to X k to X + + kk nn

nn


Power of a Statistical TestPower of a Statistical Test

== P(fail to reject H P(fail to reject Hoo when H when Hoo is false) is false)

1- 1- = = P(rejecting H P(rejecting Hoo when H when Hoo is false) is false)

1- 1- = = the power of the test the power of the test


Probability of Rejecting HProbability of Rejecting Hoo

Figure 10.6Figure 10.6

XX405405 413.16413.16

BB

Area = 1 - Area = 1 - when µ = 405when µ = 405

Area = Area = = .10 = .10

386.84386.84 400400

AA


Power of TestsPower of Tests

2.2. Power of testPower of test = = PP((ZZ > > zz11) + ) + PP((ZZ < < zz22))

1.1. DetermineDetermine

zz11 = = ZZ/2/2 - -µ - µµ - µoo

/ / nn

zz11 = - = -ZZ/2/2 - -µ - µµ - µoo

/ / nn


Power CurvePower Curve

Power curve for test of Ho versus Ha using five-step procedure

Power curve for test of Ho versus Ha using any other procedure

400 405

.1655.10

Figure 10.7

µ

1.0

1 - = P(rejecting Ho)


One Tailed Test for One Tailed Test for µµ

HHoo: µ ≥ 32.5: µ ≥ 32.5 HHaa: µ < 32.5: µ < 32.5

1.1. Define HDefine Hoo and H and Haa prior to observation prior to observation

ZZ = = XX - µ - µoo

/ / nn


3. Reject if3. Reject if

HHoo if Zif Z = < -1.645 = < -1.645XX - 32.5 - 32.5

/ / nnExample 10.4Example 10.4


One Tailed Test for One Tailed Test for µµ

5.5. Study supports the claim that the Study supports the claim that the average mileage for the Bullet is less average mileage for the Bullet is less than than 32.532.5 mpg – supports claim of mpg – supports claim of false advertisingfalse advertising

ZZ * * = = -2.70 = = -2.7030.4 - 32.530.4 - 32.5

5.5 / 505.5 / 50

4.4. The value of the test statistic isThe value of the test statistic is

Example 10.4Example 10.4


One Tailed Test for µ One Tailed Test for µ


-1.645-1.645 00ZZ

Area = .5 - .05 = .45Area = .5 - .05 = .45

Area = Area = = .05 = .05


Statistical Software Statistical Software ExampleExample

Z Z * = 1.531* = 1.531


ZZ2.332.3300

Area = .49Area = .49

Area = Area = = .01 = .01


Statistical Software ExampleStatistical Software Example




2.2. Power of test = PPower of test = P((Z Z > > zz11))


zz11 = = ZZ/2/2 - -µ - µµ - µoo

/ / nn

HHoo: µ ≤ µ: µ ≤ µoo verses verses HHaa: µ > µ: µ > µoo



2.2. Power of testPower of test = = PP((ZZ < < zz22))


zz11 = - = -ZZ/2/2 - -µ - µµ - µoo

/ / nn

HHoo: µ ≥ µ: µ ≥ µoo verses verses HHaa: µ < µ: µ < µoo


Power of TestsPower of TestsTests on a Population Mean (Tests on a Population Mean ( Known) Known)

Two-Tailed TestTwo-Tailed Test

HHoo: µ = µ: µ = µoo

HHaa: µ ≠ µ: µ ≠ µoo

rejectreject HHoo if | if |ZZ *| > *| > ZZ/2/2

One-Tailed TestOne-Tailed Test

HHoo: µ ≤ µ: µ ≤ µoo

HHaa: µ > µ: µ > µoo

rejectreject HHoo if if ZZ * > * > ZZ

HHoo: µ ≥ µ: µ ≥ µoo

HHaa: µ < µ: µ < µoo

rejectreject HHoo if if ZZ * < -* < -ZZ


Determining the p-ValueDetermining the p-Value

The p-value is the value of The p-value is the value of at at which the hypothesis test which the hypothesis test procedure changes conclusions procedure changes conclusions based on a given set of data. It is based on a given set of data. It is the largest value of the largest value of for which for which you will fail to reject Hyou will fail to reject Hoo



Figure 10.12

-1.9

6 0Z

-2.5

75

1.96

2.57

5

Z * = -2.38

Area = .025

Area =.005

Area = .025

Area = .005



Figure 10.13

0Z

-2.38 2.38

Area = p value

Area = .5 - .4913= .0087

Area = .4913 (Table A.4)


Procedure for Finding the Procedure for Finding the p-Valuep-Value

For HFor Haa: µ ≠ µ: µ ≠ µoo

p = 2p = 2 •• ( (area outside Zarea outside Z **))For HFor Haa: µ > µ: µ > µoo

p = area to the right of Zp = area to the right of Z **

For HFor Haa: µ < µ: µ < µoo

p = area to the left of Zp = area to the left of Z **



Figure 10.14

0Z

Z * = 1.53

p = area= .5 - .4370= .0630

Area = .4370


Interpreting the p-ValueInterpreting the p-Value

Classical ApproachClassical Approachreject Hreject Hoo if p-value if p-value < <

fail to reject Hfail to reject Hoo is p-value is p-value ≥ ≥ General rule of thumbGeneral rule of thumb

reject Hreject Hoo if p-value is small if p-value is small ((p p < .01)< .01)

fail to reject Hfail to reject Hoo is p-value is large is p-value is large ((p p > .1)> .1)


Interpreting the p-ValueInterpreting the p-Value

Small Small pp

Reject Reject HHoo

.01.01

Large Large pp

Fail to reject Fail to reject HHoo

pp.1.1

InconclusiveInconclusive


Another InterpretationAnother Interpretation1.1. For a two-tailed test where HFor a two-tailed test where Hoo: µ ≠ µ: µ ≠ µoo,, the p-value is the p-value is

the probability that the value of the test statistic, Zthe probability that the value of the test statistic, Z *, *, will be at least as large (in absolute value) as the will be at least as large (in absolute value) as the observed Zobserved Z *, if *, if µµ is in fact equal to is in fact equal to µµoo

2.2. For a one-tailed test where HFor a one-tailed test where Haa: µ > µ: µ > µoo,, the p-value is the p-value is

the probability that the value of the test statistic, Zthe probability that the value of the test statistic, Z *, *, will be at least as large as the observed Zwill be at least as large as the observed Z *, if *, if µµ is in is in fact equal to fact equal to µµoo

3.3. For a one-tailed test where HFor a one-tailed test where Haa: µ < µ: µ < µoo,, the p-value is the p-value is

the probability that the value of the test statistic, Zthe probability that the value of the test statistic, Z *, *, will be at least as large as the observed Zwill be at least as large as the observed Z *, if *, if µµ is in is in fact equal to fact equal to µµoo


Practical Versus StatisticalPractical Versus Statistical

Figure 10.18

ZZ * = -2.69

Area = p value= .0036 (from Table A.4)


Hypothesis Testing on the Hypothesis Testing on the Mean of a Normal Population Mean of a Normal Population

(( Unknown) Unknown)

Normal populationNormal population Population standard deviation unknownPopulation standard deviation unknown Small sampleSmall sample Student t distributionStudent t distribution

Nonparametric procedureNonparametric procedure

tt = = XX - µ - µoo

ss / / nn


Small Sample TestSmall Sample Test

(a)

Normal population

Symmetric (nonnormal) population Skewed

population

(b)

Figure 10.19


Clark Products ExampleClark Products Example1. When a question is phrased “Is there 1. When a question is phrased “Is there

evidence to indicate that ...,” what follows evidence to indicate that ...,” what follows is the alternative hypothesisis the alternative hypothesis

HHoo: µ = 10 : µ = 10 andand HHaa: µ ≠ 10: µ ≠ 10

2.2. The test statistic here isThe test statistic here is

tt = = XX - µ - µoo

ss / / nn

3.3. Using a significance level of .05 and Figure 10.20, Using a significance level of .05 and Figure 10.20, the corresponding two-tailed procedure is tothe corresponding two-tailed procedure is to

rejectreject HHoo if | if |tt| > | > tt.025,17.025,17 = 2.11 = 2.11


Clark Products ExampleClark Products Example

Figure 10.20

Z-2.11 2.11

Area = .025 Area = .025

Reject Ho here Reject Ho here



Figure 10.21

t * = 1.83

t

2.11

1.740

Area = .05

Area = .025



5. There is insufficient evidence to indicate 5. There is insufficient evidence to indicate that the average output voltage is different that the average output voltage is different from 10 voltsfrom 10 volts

4.4. The value of the test statistic isThe value of the test statistic is

tt ** = = 1.83 = = 1.83 10.331 - 1010.331 - 10

.767 / 18.767 / 18


Small-Sample Tests on aSmall-Sample Tests on aNormal Population MeanNormal Population Mean

Two-tailed testTwo-tailed test


HHaa: µ ≠ µ: µ ≠ µoo

rejectreject HHoo if | if |tt *| > *| > tt/2, /2, nn-1-1

One tail testOne tail test


HHaa: µ < µ: µ < µoo

rejectreject HHoo if if tt * < t* < t, n-1, n-1


HHaa: µ > µ: µ > µoo

rejectreject HHoo if if tt * > * > tt, , nn-1-1

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