H--TESTING contd(9th)

8/6/2019 H--TESTING contd(9th)

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11SlideSlide

Chapter 11Chapter 11Hypothesis Testing About the DifferencesHypothesis Testing About the Differences

BetweenBetween TwoTwo Population MeansPopulation Means

H0:

1 - 2 = 0Ha:

1 - 2 # 0


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22SlideSlide

Hypothesis Testing About the DifferencesHypothesis Testing About the DifferencesBetweenBetween TwoTwo Population MeansPopulation Means

LetLet QQ11 equal the mean of populationequal the mean of population 11 andand QQ22 equalequalthe mean of populationthe mean of population 22..

The difference between the two population means isThe difference between the two population means isQQ11 -- QQ22..

LetLet nn11 be the sample size of populationbe the sample size of population 11 andand nn22 thethesample size of populationsample size of population 22..

LetLet equal the mean of sampleequal the mean of sample 11 and equal theand equal the

mean of samplemean of sample 2.2.

x1

x2


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33SlideSlide

Hypothesis TestsHypothesis Tests AboutAbout QQ 11 QQ 22::WW11 andand WW22 KnownKnown

HypothesesHypotheses

1 2 0

2 21 2

1 2

( ) x x Dz

n n

W W

!

Q Q {1 2 0

:a

H D

Q Q0 1 2 0:H D Q Q e

0 1 2 0:H D

Q Q "1 2 0

:a

H D

Q Q u0 1 2 0:H D

Q Q 1 2 0

:a

LeftLeft--tailedtailed RightRight--tailedtailed TwoTwo--tailedtailed

Test Statistic (Test Statistic (Actual z ValueActual z Value))


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44SlideSlide

with a standard deviation

of $7,000 for a sample of

35 households. At the 0.01significance level can we

conclude the mean income

in Bradford is more?

T w o c i t i e s ,

Bradford and Kane

are separated onlyby the Conewango

River. There is

c o m p e t i t i o n

between the two

cities. The localpaper recently reported that

the mean household income

in Bradford is $38,000 with

a standard deviation of $6,000 for a sample of 40

househo lds . The same

article reported the mean

income in Kane is $35,000

Challenging Example 1Challenging Example 1


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55SlideSlide

Step 2

State the level of significance.The 0.01 significance level is

stated in the problem.

Step 3

Find the appropriate test

statistic. Because both

samples are more than 30, we

can use zas the test statistic.

Step 1

State the null and

alternate hypotheses.

H0: B < K

H1: B > K

Step 4

State the decision rule.

The null hypothesis is

rejected if actual z is

greater than critical z

of 2.33 or if p .01

SolutionSolution


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66SlideSlide

98.1

35

)000,7($

40

)000,6($

000,35$000,38$

22!

!z

Step 5: Compute the value of actual zand make a decision

Because the actual Z = 1.98 < critical Z = 2.33, andsince the p-value = 0.0239 > = 0.01 the decision is

to accept the null hypothesis. Thus we cannot concludethat the mean household income in Bradford is larger

then the mean household income in Kane.

Actual ZActual Z

H0: B < KH1: B > K


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77SlideSlide

WhenWhen WW11 andand WW22 are unknown, we will:are unknown, we will:

useuse tt table instead oftable instead of ZZ table.table.Assuming at least one of the sample isAssuming at least one of the sample is n < 30n < 30

use the sample standard deviationsuse the sample standard deviations ss11 andand ss22as estimates ofas estimates of WW11 andand WW22 , and, and

Hypothesis TestsHypothesis Tests AboutAbout QQ 11 QQ 22::WW11 andand WW22 Unknown and Small SampleUnknown and Small Sample


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88SlideSlide

Hypothesis TestsHypothesis Tests AboutAbout QQ 11 QQ 22::WW11 andand WW22 Unknown and Small SampleUnknown and Small Sample

HypothesesHypotheses

1 2 0

2 2

1 2

1 2

( ) x x Dt

s s

n n

!

Q Q {1 2 0

:a

H D

Q Q0 1 2 0:H D Q Q e

0 1 2 0:H D

Q Q "1 2 0

:a

H D

Q Q u0 1 2 0:H D

Q Q 1 2 0

:a

LeftLeft--tailedtailed RightRight--tailedtailed TwoTwo--tailedtailed

Test Statistic (Test Statistic (Actual tActual t))WhenWhen n < 30n < 30 andand WW isisunknown:unknown:


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1010SlideSlide

ArianaAriana Corporation wants to increase theCorporation wants to increase the

productivity ofits line workers. To do so, twoproductivity ofits line workers. To do so, two

different programs have been suggested to helpdifferent programs have been suggested to help

increaseproductivity. Twenty employees,increaseproductivity. Twenty employees,

making up a sample, have beenmaking up a sample, have been randomlyrandomly

assigned to one of the two programsassigned to one of the two programs and theirand their

output fora day's work has been recorded. Theoutput fora day's work has been recorded. The

results are given on the next slide.results are given on the next slide.

Hypothesis Tests AboutHypothesis Tests About QQ 11 !! QQ 22 WhenWhen

WW11 andand WW22U

nknown & Small SamplesU

nknown & Small SamplesChallenging ExampleChallenging Example #1#1UsingUsing SWStatSWStat++ and Data Analysis in Exceland Data Analysis in Excel


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1111SlideSlide

Example #1 ContinuedExample #1 Continued

Program A Program B150 150

130 120

120 135

180 160

145 110

185 175

220 150

190 120

180 130

175 220


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1212SlideSlide

a.a. State the null and alternative hypotheses.State the null and alternative hypotheses. b.b. Use Excel, and at 95% confidence testUse Excel, and at 95% confidence test

to determineif the means of the twoto determineif the means of the two

populations areequal, andpopulations areequal, and

c.c. Explain youranswerExplain youranswer

Example #1 ContinuedExample #1 ContinuedQuestionsQuestions


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1313SlideSlide

Excels Excels tt--Test: Two Sample Assuming UnequalTest: Two Sample Assuming Unequal

Variances ToolVariances Tool

Step 1Step 1 Select theSelect the ToolsTools menumenu

Step 2Step 2 Choose theChoose the Data AnalysisData Analysis optionoption

Step 3Step 3 ChooseChoose tt--Test:TwoSample AssumingTest:TwoSample Assuming

Unequal VariancesUnequal Variances from the list offrom the list of

Analysis ToolsAnalysis Tools

continued continued

Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 Unknown & Small SamplesUnknown & Small Samples

U

sing Data Analysis in ExcelU

sing Data Analysis in Excel


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1414SlideSlide

Excel Dial

ogBo

xExcel Dial

ogBo

x

Solution to #1 Using Data Analysis in ExcelSolution to #1 Using Data Analysis in Excel


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1515SlideSlide

Solution to #1 Using Data Analysis in ExcelSolution to #1 Using Data Analysis in Excel

H0: Q1 = Q2

Ha: Q1 # Q2


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1616SlideSlide

SWStat+ t-Test: TwoSample AssumingUnequal Variances Tool

Step 1Step 1 CreateCreate DataData areaarea

Step 2Step 2 ChooseChoose StatisticsStatistics, and then choose, and then choosethethe Intervals and TestsIntervals and Tests optionoption

Step 3Step 3 FromFrom TypeType, Choose, Choose TwoSamples,TwoSamples,

Different Means, Unequal Variances,Different Means, Unequal Variances, tt--TestTest

continued continued

Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 Unknown & Small SamplesUnknown & Small Samples


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1717SlideSlide

Solution to #1 UsingSolution to #1 Using SWStatSWStat++

(Creating Data Area)(Creating Data Area)

Data Area


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1818SlideSlide


(Selecting Intervals and Tests Option)(Selecting Intervals and Tests Option)


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1919SlideSlide


(Results)(Results)

H0: Q1 = Q2

Ha: Q1 # Q2


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2020SlideSlide

Information regarding theACT scores of

samples of twenty two students in two

different majors at CSU is given in next slide.

Questions: (a) Use Excel, and at 95%

confidence test to determine whether thereis

a significant differencein the means of the two

populations, and (b) explain youranswer.

Hypothesis Tests AboutHypothesis Tests About QQ 11 !! QQ 22 WhenWhenWW11 andand WW22 UnknownUnknown

Challenging ExampleChallenging Example #2#2


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2121SlideSlide

DATADATA

H0: Q1 = Q2Ha: Q1 # Q2

Hypothesis TestHypothesis Test


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2222SlideSlide

Solution to #2Using Data Analysis in ExcelSolution to #2Using Data Analysis in Excel


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2323SlideSlide

Solution to #2UsingSolution to #2Using SWStat+SWStat+(Results)(Results)

TwoSamples, Different Means, Unequal Variances, t TestTwoSamples, Different Means, Unequal Variances, t Test

H0: Q1 = Q2

Ha: Q1 # Q2


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2424SlideSlide

If samples are large (n130 and n230) and also ifpopulation standard deviations (1 and 2) aregiven then you can use either Data Analysis or

SWStat+

in Excel to solve the problem.

Note that in this case, the option that you use inData Analysis is z-Test: TwoSample for Meansand the option that you use in SWStat+ is Two

Sample, diff Variances/SDs: F See next example.

Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 areare KnownKnown


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2525SlideSlide

In a test of driving distance using a mechanicalIn a test of driving distance using a mechanical

driving device, a sample of Par golf balls wasdriving device, a sample of Par golf balls was

compared with a sample of golf balls made by Rap,compared with a sample of golf balls made by Rap,

Ltd., a competitor. The sample statistics appear on theLtd., a competitor. The sample statistics appear on the

next slide.next slide.

Par, Inc. is a manufacturerPar, Inc. is a manufacturer

of golf equipment and hasof golf equipment and has

developed a new golf balldeveloped a new golf ball

that has been designed tothat has been designed toprovide extra distance.provide extra distance.

Example:

Par, Inc.

Example:

Par, Inc.

Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 areare KnownKnown


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2626SlideSlide

Example:

Par, Inc.

Example:

Par, Inc.

Hypothesis Testing ofHypothesis Testing of QQ11 -- QQ22::WW11 andand WW22 KnownKnown

Sample SizeSample Size

Sample MeanSample Mean

Sample #1Sample #1Par, Inc.Par, Inc.

Sample #2Sample #2Rap, Ltd.Rap, Ltd.

120 balls120 balls 80 balls80 balls

275 yards 258 yards275 yards 258 yards

Given that the two population standard deviations areGiven that the two population standard deviations are

known as:known as: 11 = 15= 15 yards andyards and 22 = 20= 20 yards, thenyards, then

Can we conclude, using = 0.01, that the mean

driving distance of Par, Inc. golf balls is greater than

the mean driving distance of Rap, Ltd. golf balls?


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2727SlideSlide

HH00:: QQ11 -- QQ22 0> 0

where:where:QQ11 = mean distance for the population= mean distance for the population

of Par, Inc. golf ballsof Par, Inc. golf ballsQQ

22= mean distance for the population= mean distance for the population

of Rap, Ltd. golf ballsof Rap, Ltd. golf balls

The hypotheses is:The hypotheses is:

pp Value and Critical Value ApproachesValue and Critical Value Approaches

The level of significance is:The level of significance is: EE = .01= .01

Hypothesis Testing ofHypothesis Testing of QQ11 -- QQ22 WhenWhenWW11 andand WW22 are Knownare Known


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2828SlideSlide

Excels Excels zz--Test: Two Sample for Means ToolTest: Two Sample for Means Tool

Step 1Step 1 After data entry, thenAfter data entry, then select theselect the ToolsTools menumenu

Step 2Step 2 Choose theChoose the Data AnalysisData Analysis optionoption

Step 3Step 3 ChooseChoose zz--Test:TwoSample for MeansTest:TwoSample for Means

from the list of Analysis Toolsfrom the list of Analysis Tools

continued continued

Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 are Knownare Known


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2929SlideSlide

Excel Dialog BoxExcel Dialog Box


HH00:: QQ11 -- QQ22 0> 0

HH00:: QQ11 QQ22

HHaa:: QQ11 >> QQ22OROR


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3030SlideSlide

Excel Value WorksheetExcel Value Worksheet

Note: Rows 14Note: Rows 14--121 are not shown.121 are not shown.

A B C D E F

1 Par Rap

2 255 266 z-Test: T ple f rMe s

3 270 238

4 294 243 r, Inc. p, t .

5 245 277 Me n 235 218

6 300 275 Known V riance 225 400

7 262 244 Obser ations 120 80

8 281 239 Hypot esize Mean Difference 09 257 242 z 6.483545607

10 268 280 P(Z< z) one-tail 4.47959E-1111 295 261 z ritical one-tail 2.326347874

12 249 276 P(Z< z) two-tail 8.95919E-11

13 291 241 z ritical two-tail 2.575829304



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3131SlideSlide

BecauseBecauseppvalue = 0value = 0 > critical z = 2.33critical z = 2.33wewe rejectreject HH

00. In other words:. In other words:

ConclusionConclusion

At the0.01 level of significance, the sampleevidence

indicates that the mean driving distance of Par, Inc. golf

balls ( 1) is greater than the mean driving distance of

Rap, Ltd. golf balls (2). In otherwords

HH00:: QQ11 -- QQ22 0> 0HH

00:: QQ11 QQ22

HHaa:: QQ11 >> QQ22OROR


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End of Chapter11

DANCERDANCER

H--TESTING contd(9th)

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