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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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Solution to #1 UsingSolution to #1 Using SWStatSWStat++
(Creating Data Area)(Creating Data Area)
Data Area
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1818SlideSlide
Solution to #1 UsingSolution to #1 Using SWStatSWStat++
(Selecting Intervals and Tests Option)(Selecting Intervals and Tests Option)
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Solution to #1 UsingSolution to #1 Using SWStatSWStat++
(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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Solution to #2Using Data Analysis in ExcelSolution to #2Using Data Analysis in Excel
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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
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2929SlideSlide
Excel Dialog BoxExcel Dialog Box
Hypothesis Tests AboutHypothesis Tests About QQ 11 QQ 22 WhenWhenWW11 andand WW22 are Knownare Known
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