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2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Inferences Based on a Single Sample: Inferences Based on a Single Sample: Tests of HypothesisTests of Hypothesis
8 8 -- 11
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Learning ObjectivesLearning Objectives
1.1. Distinguish Types of Hypotheses Distinguish Types of Hypotheses
2.2. Describe Hypothesis Testing ProcessDescribe Hypothesis Testing Process
3.3. Explain pExplain p--Value ConceptValue Concept
8 8 -- 22
3.3. Explain pExplain p--Value ConceptValue Concept
4.4. Solve Hypothesis Testing Problems Solve Hypothesis Testing Problems Based on a Single SampleBased on a Single Sample
5.5. Explain Power of a TestExplain Power of a Test
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Statistical MethodsStatistical Methods
StatisticalMethods
8 8 -- 33
DescriptiveStatistics
InferentialStatistics
EstimationHypothesis
Testing
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis Testing ConceptsHypothesis Testing Concepts
8 8 -- 44
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis TestingHypothesis Testing
8 8 -- 55
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis TestingHypothesis Testing
PopulationPopulation
8 8 -- 66
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis TestingHypothesis Testing
PopulationPopulation
I believe the population mean age is 50 (hypothesis).
8 8 -- 77
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis TestingHypothesis Testing
PopulationPopulation
I believe the population mean age is 50 (hypothesis).
8 8 -- 88
MeanMeanX X = 20= 20
Random Random samplesample
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis TestingHypothesis Testing
PopulationPopulation
I believe the population mean age is 50 (hypothesis).
Reject hypothesis! Not close.
8 8 -- 99
MeanMeanX X = 20= 20
Random Random samplesample
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Whats a Hypothesis?Whats a Hypothesis?
1.1. A Belief about a A Belief about a Population ParameterPopulation Parameter
Parameter Is Parameter Is
I believe the mean GPA I believe the mean GPA of this class is 3.5!of this class is 3.5!
8 8 -- 1010
Parameter Is Parameter Is PopulationPopulation Mean, Mean, Proportion, VarianceProportion, Variance
Must Be StatedMust Be StatedBeforeBefore AnalysisAnalysis
1984-1994 T/Maker Co.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Null HypothesisNull Hypothesis
1.1. What Is TestedWhat Is Tested
2.2. Has Serious Outcome If Incorrect Has Serious Outcome If Incorrect Decision MadeDecision Made
8 8 -- 1111
3.3. Designated HDesignated H00 (Pronounced H(Pronounced H--noughtnought))
4.4. Specified as HSpecified as H00: : Some Numeric Value Some Numeric Value Specified with = Sign Specified with = Sign , or , or Example, HExample, H00: : 33
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Alternative HypothesisAlternative Hypothesis
1.1. Opposite of Null HypothesisOpposite of Null Hypothesis
2.2. Always Has Inequality Sign:Always Has Inequality Sign: ,,, or , or 3.3. Designated HDesignated H
8 8 -- 1212
3.3. Designated HDesignated Haa4.4. Specified HSpecified Haa: : < Some Value< Some Value
Example, HExample, Haa: : < 3< 3 will lead towill lead to twotwo--sided testssided tests will lead to one will lead to one--sided testssided tests
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Identifying HypothesesIdentifying HypothesesStepsSteps
1.1. Example Problem: Test That the Example Problem: Test That the Population Mean Is Not 3Population Mean Is Not 3
2.2. StepsSteps
8 8 -- 1313
2.2. StepsSteps State the Question Statistically (State the Question Statistically ( 3)3) State the Opposite Statistically (State the Opposite Statistically ( = 3)= 3)
Must Be Mutually Exclusive & ExhaustiveMust Be Mutually Exclusive & Exhaustive
Select the Alternative Hypothesis (Select the Alternative Hypothesis ( 3)3) Has the Has the , , SignSign
State the Null Hypothesis (State the Null Hypothesis ( = 3)= 3)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
State the question statistically: State the question statistically: = 12= 12
Is the population average amount of TV Is the population average amount of TV viewing 12 hours?viewing 12 hours?
What Are the Hypotheses?What Are the Hypotheses?
8 8 -- 1414
State the opposite statistically: State the opposite statistically: 12 12 Select the alternative hypothesis: Select the alternative hypothesis: HHaa: : 1212State the null hypothesis: State the null hypothesis: HH00: : = 12= 12
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
State the question statistically: State the question statistically: 1212
Is the population average amount of TV Is the population average amount of TV viewing viewing differentdifferent from 12 hours?from 12 hours?
What Are the Hypotheses?What Are the Hypotheses?
8 8 -- 1515
State the opposite statistically: State the opposite statistically: = 12= 12Select the alternative hypothesis: Select the alternative hypothesis: HHaa: : 1212State the null hypothesis: State the null hypothesis: HH00: : = 12= 12
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
State the question statistically: State the question statistically: 2020
Is the average cost per hat less than or Is the average cost per hat less than or equal to P20?equal to P20?
What Are the Hypotheses?What Are the Hypotheses?
8 8 -- 1616
State the opposite statistically: State the opposite statistically: 2020Select the alternative hypothesis: Select the alternative hypothesis: HHaa: : 2020State the null hypothesis: State the null hypothesis: HH00: : 2020
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
State the question statistically: State the question statistically: 2525
Is the average amount spent in the Is the average amount spent in the bookstore greater than P25?bookstore greater than P25?
What Are the Hypotheses?What Are the Hypotheses?
8 8 -- 1717
State the opposite statistically: State the opposite statistically: 25 25 Select the alternative hypothesis: Select the alternative hypothesis: HHaa: : 2525State the null hypothesis: State the null hypothesis: HH00: : 2525
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Basic IdeaBasic Idea
8 8 -- 1818
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Basic IdeaBasic Idea
Sampling DistributionSampling Distribution
8 8 -- 1919
Sample Mean = 50HH00HH00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Basic IdeaBasic Idea
Sampling DistributionSampling Distribution
It is unlikely It is unlikely that we would that we would get a sample get a sample
8 8 -- 2020
Sample Mean = 50
mean of this mean of this value ...value ...
20202020HH00HH00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Basic IdeaBasic Idea
Sampling DistributionSampling Distribution
It is unlikely It is unlikely that we would that we would get a sample get a sample
8 8 -- 2121
Sample Mean = 50
mean of this mean of this value ...value ...
... if in fact this were... if in fact this werethe population meanthe population mean
20202020HH00HH00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Basic IdeaBasic Idea
Sampling DistributionSampling Distribution
It is unlikely It is unlikely that we would that we would get a sample get a sample
... therefore, ... therefore, we reject the we reject the
8 8 -- 2222
Sample Mean = 50
mean of this mean of this value ...value ...
... if in fact this were... if in fact this werethe population meanthe population mean
hypothesis hypothesis that that = 50.= 50.
20202020HH00HH00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Level of SignificanceLevel of Significance
1.1. ProbabilityProbability
2.2. Defines Unlikely Values of Sample Defines Unlikely Values of Sample Statistic if Null Hypothesis Is TrueStatistic if Null Hypothesis Is True
8 8 -- 2323
Statistic if Null Hypothesis Is TrueStatistic if Null Hypothesis Is True Called Rejection Region of Sampling Called Rejection Region of Sampling
DistributionDistribution
3.3. Designated Designated (alpha)(alpha) Typical Values Are .01, .05, .10Typical Values Are .01, .05, .10
4.4. Selected by Researcher at StartSelected by Researcher at Start
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Region Rejection Region (One(One--Tail Test) Tail Test)
8 8 -- 2424
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Region Rejection Region (One(One--Tail Test) Tail Test)
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 2525
HoValueCritical
Value
Sample Statistic
NonrejectionRegion
1 1 --
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Region Rejection Region (One(One--Tail Test) Tail Test)
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 2626
HoValueCritical
Value
Sample Statistic
NonrejectionRegion
1 1 --
Observed sample statisticObserved sample statistic
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Region Rejection Region (One(One--Tail Test) Tail Test)
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 2727
HoValueCritical
Value
Sample Statistic
NonrejectionRegion
1 1 --
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Regions Rejection Regions (Two(Two--Tailed Test) Tailed Test)
8 8 -- 2828
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Regions Rejection Regions (Two(Two--Tailed Test) Tailed Test)
RejectionRegion
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 2929
HoValue Critical
ValueCriticalValue
1/2 1/2
Sample Statistic
NonrejectionRegion
1 1 --
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Regions Rejection Regions (Two(Two--Tailed Test) Tailed Test)
RejectionRegion
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 3030
HoValue Critical
ValueCriticalValue
1/2 1/2
Sample Statistic
NonrejectionRegion
1 1 --
Observed sample statisticObserved sample statistic
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Regions Rejection Regions (Two(Two--Tailed Test) Tailed Test)
RejectionRegion
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 3131
HoValue Critical
ValueCriticalValue
1/2 1/2
Sample Statistic
NonrejectionRegion
1 1 --
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Rejection Regions Rejection Regions (Two(Two--Tailed Test) Tailed Test)
RejectionRegion
RejectionRegion
Sampling DistributionSampling DistributionLevel of ConfidenceLevel of Confidence
8 8 -- 3232
HoValue Critical
ValueCriticalValue
1/2 1/2
Sample Statistic
NonrejectionRegion
1 1 --
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Hypothesis Testing StepsHypothesis Testing Steps
8 8 -- 3333
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
HH00 Testing StepsTesting Steps
8 8 -- 3434
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
HH00 Testing StepsTesting Steps
State HState H00
State HState Haa
Choose Choose
8 8 -- 3535
Choose Choose
Choose Choose nn
Choose testChoose test
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
HH00 Testing StepsTesting Steps
Set up critical valuesSet up critical values
Collect dataCollect data
Compute test statisticCompute test statistic
State HState H00
State HState Haa
Choose Choose
8 8 -- 3636
Compute test statisticCompute test statistic
Make statistical decisionMake statistical decision
Express decisionExpress decision
Choose Choose
Choose Choose nn
Choose testChoose test
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
One Population TestsOne Population Tests
OnePopulation
8 8 -- 3737
Z Test(1 & 2tail)
t Test(1 & 2tail)
Z Test(1 & 2tail)
Mean Proportion Variance
2 Test(1 & 2tail)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test of Mean (of Mean ( Known)Known)
8 8 -- 3838
of Mean (of Mean ( Known)Known)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
One Population TestsOne Population Tests
OnePopulation
8 8 -- 3939
Z Test(1 & 2tail)
t Test(1 & 2tail)
Z Test(1 & 2tail)
Mean Proportion Variance
2 Test(1 & 2tail)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test for Mean (for Mean ( Known)Known)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribution (nn 30)30)
8 8 -- 4040
Normal Distribution (Normal Distribution (nn 30)30)2.2. Alternative Hypothesis Has Alternative Hypothesis Has SignSign
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test for Mean (for Mean ( Known)Known)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribution (nn 30)30)
8 8 -- 4141
Normal Distribution (Normal Distribution (nn 30)30)2.2. Alternative Hypothesis Has Alternative Hypothesis Has SignSign3.3. ZZ--Test StatisticTest Statistic
ZX X
n
x
x
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z TestTailed Z TestExample Example
Does an average box of Does an average box of cereal contain cereal contain 368368 grams grams of cereal? A random of cereal? A random sample of sample of 2525 boxes boxes
8 8 -- 4242
sample of sample of 2525 boxes boxes showedshowedX = 372.5X = 372.5. The . The company has specified company has specified to be to be 2525 grams. Test at grams. Test at the the .05.05 level.level. 368 gm.368 gm.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: :
HHaa: :
nn
Test Statistic: Test Statistic:
8 8 -- 4343
nn Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 nn
Test Statistic: Test Statistic:
8 8 -- 4444
nn Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 .05.05nn 2525
Test Statistic: Test Statistic:
8 8 -- 4545
nn 2525Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 .05.05nn 2525
Test Statistic: Test Statistic:
8 8 -- 4646
nn 2525Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 .05.05nn 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 4747
nn 2525Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 25
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 .05.05nn 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 4848
nn 2525Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 25
Do not reject at Do not reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 .05.05nn 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 4949
nn 2525Critical Value(s):Critical Value(s): Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 25
Do not reject at Do not reject at = .05= .05
No evidence No evidence average is not 368average is not 368
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Thinking ChallengeThinking Challenge
Youre a Q/C inspector. You want to Youre a Q/C inspector. You want to find out if a new machine is making find out if a new machine is making electrical cords to customer electrical cords to customer specification: specification: averageaverage breaking breaking
8 8 -- 5050
specification: specification: averageaverage breaking breaking strength of strength of 7070 lb. with lb. with = 3.5= 3.5 lb. lb. You take a sample of You take a sample of 3636 cords & cords & compute a sample mean of compute a sample mean of 69.769.7 lb. lb. At the At the .05.05 level, is there evidence level, is there evidence that the machine is that the machine is notnot meeting the meeting the average breaking strength?average breaking strength?
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: :
HHaa: :
= = nn = =
Test Statistic: Test Statistic:
8 8 -- 5151
nn = = Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = nn ==
Test Statistic: Test Statistic:
8 8 -- 5252
nn ==Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = .05.05nn = = 3636
Test Statistic: Test Statistic:
8 8 -- 5353
nn = = 3636Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = .05.05nn = = 3636
Test Statistic: Test Statistic:
8 8 -- 5454
nn = = 3636Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = .05.05nn = = 3636
Test Statistic: Test Statistic:
ZX
n
69 7 70
3 536
51.
..
8 8 -- 5555
nn = = 3636Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 36
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = .05.05nn = = 3636
Test Statistic: Test Statistic:
ZX
n
69 7 70
3 536
51.
..
8 8 -- 5656
nn = = 3636Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 36
Do not reject at Do not reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 70= 70HHaa: : 7070 = = .05.05nn = = 3636
Test Statistic: Test Statistic:
ZX
n
69 7 70
3 536
51.
..
8 8 -- 5757
nn = = 3636Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n 36
Do not reject at Do not reject at = .05= .05
No evidence No evidence average is not 70average is not 70
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test of Mean (of Mean ( Known)Known)
8 8 -- 5858
of Mean (of Mean ( Known)Known)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test for Mean (for Mean ( Known)Known)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribution (nn 30)30)
8 8 -- 5959
Normal Distribution (Normal Distribution (nn 30)30)2.2. Alternative Hypothesis Has < or > SignAlternative Hypothesis Has < or > Sign
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test for Mean (for Mean ( Known)Known)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribution (nn 30)30)
8 8 -- 6060
Normal Distribution (Normal Distribution (nn 30)30)2.2. Alternative Hypothesis Has Alternative Hypothesis Has or > Signor > Sign3.3. ZZ--test Statistictest Statistic
ZX X
n
x
x
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test for Mean Hypothesesfor Mean Hypotheses
8 8 -- 6161
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Reject H0
OneOne--Tailed Z Test Tailed Z Test for Mean Hypothesesfor Mean Hypotheses
HH00::==0 H0 Haa: :
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Reject H0Reject H0
OneOne--Tailed Z Test Tailed Z Test for Mean Hypothesesfor Mean Hypotheses
HH00::==0 H0 Haa: : > 00
8 8 -- 6363
Z0
Z0
Must be Must be significantlysignificantlybelow below
Small values satisfy Small values satisfy HH0 0 . Dont reject!. Dont reject!
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Finding Critical ZFinding Critical Z
8 8 -- 6464
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
= 1
OneOne--Tailed Z Test Tailed Z Test Finding Critical ZFinding Critical Z
What Is Z given What Is Z given = .025?= .025?
8 8 -- 6565
Z0
= .025= .025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
= 1
OneOne--Tailed Z Test Tailed Z Test Finding Critical ZFinding Critical Z
.500 .500 -- .025.025
What Is Z given What Is Z given = .025?= .025?
8 8 -- 6666
Z0
-- .025.025.475.475
= .025= .025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Z .05 .07 = 1
OneOne--Tailed Z Test Tailed Z Test Finding Critical ZFinding Critical Z
.500 .500 -- .025.025 .06
Standardized Normal Standardized Normal Probability Table (Portion)Probability Table (Portion)
What Is Z given What Is Z given = .025?= .025?
8 8 -- 6767
1.6 .4505 .4515 .4525
1.7 .4599 .4608 .4616
1.8 .4678 .4686 .4693
.4744 .4756
Z0
-- .025.025.475.475
1.9 .4750.4750
= .025= .025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Z .05 .07 = 1
OneOne--Tailed Z Test Tailed Z Test Finding Critical ZFinding Critical Z
.500 .500 -- .025.025 .06.06
Standardized Normal Standardized Normal Probability Table (Portion)Probability Table (Portion)
What Is Z given What Is Z given = .025?= .025?
8 8 -- 6868
1.6 .4505 .4515 .4525
1.7 .4599 .4608 .4616
1.8 .4678 .4686 .4693
.4744 .4756
Z0 1.96
-- .025.025.475.475
1.91.9 .4750
= .025= .025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z TestTailed Z TestExample Example
Does an average box of Does an average box of cereal contain cereal contain more thanmore than368368 grams of cereal? A grams of cereal? A random sample of random sample of 2525
8 8 -- 6969
random sample of random sample of 2525boxes showedboxes showedX = 372.5X = 372.5. . The company has The company has specified specified to be to be 2525grams. Test at the grams. Test at the .05.05level.level.
368 gm.368 gm.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: :
HHaa: :
= = n n = =
Test Statistic: Test Statistic:
8 8 -- 7070
n n = =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = n n = =
Test Statistic: Test Statistic:
8 8 -- 7171
n n = =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = .05.05n n = = 2525
Test Statistic: Test Statistic:
8 8 -- 7272
n n = = 2525
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = .05.05n n = = 2525
Test Statistic: Test Statistic:
8 8 -- 7373
n n = = 2525
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.645
.05
Reject
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = .05.05n n = = 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 7474
n n = = 2525
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.645
.05
Reject
n 25
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = .05.05n n = = 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 7575
n n = = 2525
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.645
.05
Reject
n 25
Do not reject at Do not reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test SolutionSolution
HH00: : = 368= 368HHaa: : > 368> 368 = = .05.05n n = = 2525
Test Statistic: Test Statistic:
ZX
n
372 5 368
1525
150.
.
8 8 -- 7676
n n = = 2525
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.645
.05
Reject
n 25
Do not reject at Do not reject at = .05= .05
No evidence average No evidence average is more than 368is more than 368
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Thinking ChallengeThinking Challenge
Youre an analyst for Ford. You Youre an analyst for Ford. You want to find out if the average want to find out if the average miles per gallon of Escorts is at miles per gallon of Escorts is at least 32 mpg. Similar models least 32 mpg. Similar models
8 8 -- 7777
least 32 mpg. Similar models least 32 mpg. Similar models have a standard deviation of have a standard deviation of 3.83.8mpg. You take a sample of mpg. You take a sample of 6060Escorts & compute a sample Escorts & compute a sample mean of mean of 30.730.7 mpg. At the mpg. At the .01.01level, is there evidence that the level, is there evidence that the miles per gallon is miles per gallon is at leastat least 3232??
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: :
HHaa: :
= = nn ==
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
8 8 -- 7878
nn ==
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 = = nn ==
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
8 8 -- 7979
nn ==
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 == .01.01nn = = 6060
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
8 8 -- 8080
nn = = 6060
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 = .01= .01nn = 60= 60
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
8 8 -- 8181
nn = 60= 60
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
Z0-2.33
.01
Reject
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 = .01= .01nn = 60= 60
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
ZX
n
30 7 32
3 860
2 65.
..
8 8 -- 8282
nn = 60= 60
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
Z0-2.33
.01
Reject
n 60
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 = .01= .01nn = 60= 60
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
ZX
n
30 7 32
3 860
2 65.
..
8 8 -- 8383
nn = 60= 60
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
Z0-2.33
.01
Reject
n 60
Reject at Reject at = .01= .01
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test Solution*Solution*
HH00: : = 32= 32HHaa: : < 32< 32 = .01= .01nn = 60= 60
Test Statistic: Test Statistic: Test Statistic: Test Statistic:
ZX
n
30 7 32
3 860
2 65.
..
8 8 -- 8484
nn = 60= 60
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Decision:Decision:
Conclusion:Conclusion:
Z0-2.33
.01
Reject
n 60
Reject at Reject at = .01= .01
There is evidence There is evidence average is less than 32average is less than 32
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Decision Making RisksDecision Making Risks
8 8 -- 8585
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Errors in Errors in Making DecisionMaking Decision
1.1. Type I ErrorType I Error Reject True Null HypothesisReject True Null Hypothesis Has Serious ConsequencesHas Serious Consequences
Probability of Type I Error Is Probability of Type I Error Is (Alpha)(Alpha)
8 8 -- 8686
Probability of Type I Error Is Probability of Type I Error Is (Alpha)(Alpha)Called Level of SignificanceCalled Level of Significance
2.2. Type II ErrorType II Error Do Not Reject False Null HypothesisDo Not Reject False Null Hypothesis Probability of Type II Error Is Probability of Type II Error Is (Beta)(Beta)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Jury Trial H0 Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0
Decision ResultsDecision Results
HH00: Innocent: Innocent
8 8 -- 8787
Verdict Innocent Guilty Decision H0 True H0False
Innocent Correct ErrorDo NotReject
H0
1 - Type IIError
()
Guilty Error Correct RejectH0
Type IError () Power(1 - )
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Jury Trial H0 Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0
Decision ResultsDecision Results
HH00: Innocent: Innocent
8 8 -- 8888
Verdict Innocent Guilty Decision H0 True H0False
Innocent Correct Error AcceptH0
1 - Type IIError
()
Guilty Error Correct RejectH0
Type IError () Power(1 - )
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
& & Have an Have an Inverse RelationshipInverse Relationship
You cant reduce both errors simultaneously!
8 8 -- 8989
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Factors Affecting Factors Affecting 1.1. True Value of Population ParameterTrue Value of Population Parameter
Increases When Difference With Hypothesized Increases When Difference With Hypothesized Parameter DecreasesParameter Decreases
2.2. Significance Level, Significance Level,
8 8 -- 9090
2.2. Significance Level, Significance Level, Increases When Increases When DecreasesDecreases
3.3. Population Standard Deviation, Population Standard Deviation, Increases When Increases When Increases Increases
4.4. Sample Size, Sample Size, nn
Increases When Increases When nn DecreasesDecreases
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Exercise 8.15Exercise 8.15
1000 subjects1000 subjects
500 told truth, 500 lied500 told truth, 500 lied
Lie detector says Lie detector says
8 8 -- 9191
185 truth tellers were liars185 truth tellers were liars
120 liars were truth tellers120 liars were truth tellers
Ho: truth tellerHo: truth teller
a)a) What is a typeWhat is a type--I error? TypeI error? Type--II error?II error?
b)b) What is Pr(typeWhat is Pr(type--I error)? Pr(typeI error)? Pr(type--II error)?II error)?
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Observed Significance Observed Significance Levels: pLevels: p--ValuesValues
8 8 -- 9292
Levels: pLevels: p--ValuesValues
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
pp--ValueValue
1.1. Probability of Obtaining a Test Statistic Probability of Obtaining a Test Statistic More Extreme (More Extreme (or or than Actual than Actual Sample Value Given HSample Value Given H00 Is True Is True
8 8 -- 9393
2.2. Called Observed Level of SignificanceCalled Observed Level of Significance Smallest Value of Smallest Value of HH00 Can Be RejectedCan Be Rejected
3.3. Used to Make Rejection DecisionUsed to Make Rejection Decision If pIf p--Value Value , Do Not Reject H, Do Not Reject H00 If pIf p--Value < Value < , Reject H, Reject H00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value Example Value Example
Does an average box of Does an average box of cereal contain cereal contain 368368 grams grams of cereal? A random of cereal? A random sample of sample of 2525 boxes boxes
8 8 -- 9494
sample of sample of 2525 boxes boxes showedshowedX = 372.5X = 372.5. The . The company has specified company has specified to be to be 2525 grams. Find the grams. Find the pp--Value.Value. 368 gm.368 gm.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
8 8 -- 9595
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
ZX
n
372 5 368
1525
150.
.
8 8 -- 9696
Z0 1.50-1.50
Z value of sample Z value of sample statistic (observed)statistic (observed)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
pp--value is P(Z value is P(Z --1.50 or Z 1.50 or Z 1.50)1.50)
8 8 -- 9797
Z0 1.50-1.50
Z value of sample Z value of sample statistic (observed)statistic (observed)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
1/2 p-Value1/2 p-Value
pp--value is P(Z value is P(Z --1.50 or Z 1.50 or Z 1.50)1.50)
8 8 -- 9898
Z0 1.50-1.50
Z value of sample Z value of sample statistic (observed)statistic (observed)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
1/2 p-Value1/2 p-Value
pp--value is P(Z value is P(Z --1.50 or Z 1.50 or Z 1.50)1.50)
8 8 -- 9999
Z0 1.50-1.50
Z value of sample Z value of sample statistic (observed)statistic (observed)
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
1/2 p-Value1/2 p-Value.5000.5000
pp--value is P(Z value is P(Z --1.50 or Z 1.50 or Z 1.50)1.50)
8 8 -- 100100
Z0 1.50-1.50
Z value of sample Z value of sample statistic (observed)statistic (observed)
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
.5000.5000-- .4332.4332
.0668.0668
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
1/2 p-Value.0668
1/2 p-Value.0668
pp--value is P(Z value is P(Z --1.50 or Z 1.50 or Z 1.50) = 1.50) = .1336.1336
.5000.5000
8 8 -- 101101
Z0 1.50-1.50
.0668.0668
Z value of sample Z value of sample statisticstatistic
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
.5000.5000-- .4332.4332
.0668.0668
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
RejectReject
1/2 p1/2 p--Value = .0668Value = .06681/2 p1/2 p--Value = .0668Value = .0668
8 8 -- 102102
0 1.50-1.50 Z
RejectReject
1/2 1/2 = .025= .0251/2 1/2 = .025= .025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
RejectReject
(p(p--Value = .1336) Value = .1336) (( = .05). = .05). Do not reject.Do not reject.
1/2 p1/2 p--Value = .0668Value = .06681/2 p1/2 p--Value = .0668Value = .0668
8 8 -- 103103
0 1.50-1.50 Z
RejectReject
1/2 1/2 = .025= .0251/2 1/2 = .025= .025
Test statistic is in Do not reject regionTest statistic is in Do not reject region
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value Example Value Example
Does an average box of Does an average box of cereal contain cereal contain more thanmore than368368 grams of cereal? A grams of cereal? A random sample of random sample of 2525
8 8 -- 104104
random sample of random sample of 2525boxes showedboxes showedX = 372.5X = 372.5. . The company has The company has specified specified to be to be 2525grams. Find the pgrams. Find the p--Value.Value. 368 gm.368 gm.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
8 8 -- 105105
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
ZX
n
372 5 368
1525
150.
.
8 8 -- 106106
Z0 1.50
Z value of sample Z value of sample statisticstatistic
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
p-ValueUse Use
pp--Value is P(Z Value is P(Z 1.50) 1.50)
8 8 -- 107107
Z0 1.50
p-ValueUse Use alternative alternative hypothesis hypothesis to find to find directiondirection
Z value of sample Z value of sample statisticstatistic
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
p-ValueUse Use
pp--Value is P(Z Value is P(Z 1.50) 1.50)
8 8 -- 108108
Z0 1.50
p-ValueUse Use alternative alternative hypothesis hypothesis to find to find directiondirection
Z value of sample Z value of sample statisticstatistic
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
p-ValueUse Use
pp--Value is P(Z Value is P(Z 1.50) 1.50)
8 8 -- 109109
Z0 1.50
p-ValueUse Use alternative alternative hypothesis hypothesis to find to find directiondirection
Z value of sample Z value of sample statisticstatistic
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
.5000.5000-- .4332.4332
.0668.0668
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
p-Value.0668Use Use
pp--Value is P(Z Value is P(Z 1.50) = .06681.50) = .0668
8 8 -- 110110
Z0 1.50
.0668
Z value of sample Z value of sample statisticstatistic
From Z table: From Z table: lookup 1.50lookup 1.50
.4332.4332
Use Use alternative alternative hypothesis hypothesis to find to find directiondirection
.5000.5000-- .4332.4332
.0668.0668
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
Reject
pp--Value = .0668Value = .0668
8 8 -- 111111
0 1.50 Z
Reject
= .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed Z Test Tailed Z Test pp--Value SolutionValue Solution
Reject
(p(p--Value = .0668) Value = .0668) (( = .05). = .05). Do not reject.Do not reject.
pp--Value = .0668Value = .0668
8 8 -- 112112
0 1.50 Z
Reject
= .05= .05
Test statistic is in Do not reject regionTest statistic is in Do not reject region
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
pp--Value Value Thinking ChallengeThinking Challenge
Youre an analyst for Ford. You Youre an analyst for Ford. You want to find out if the average want to find out if the average miles per gallon of Escorts is miles per gallon of Escorts is at at least 32 least 32 mpg. Similar models mpg. Similar models
8 8 -- 113113
least 32 least 32 mpg. Similar models mpg. Similar models have a standard deviation of have a standard deviation of 3.83.8mpg. You take a sample of mpg. You take a sample of 6060Escorts & compute a sample Escorts & compute a sample mean of mean of 30.730.7 mpg. What is the mpg. What is the value of the observed level of value of the observed level of significance (significance (pp--ValueValue)?)?
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
pp--Value Value Solution*Solution*
p-Value.004Use Use .5000.5000
pp--Value is P(Z Value is P(Z --2.65) = .004.2.65) = .004.pp--Value < (Value < ( = .01). Reject H= .01). Reject H00..
8 8 -- 114114
Z0-2.65
.004
Z value of Z value of sample statisticsample statistic
From Z table: From Z table: lookup 2.65lookup 2.65
.4960.4960
Use Use alternative alternative hypothesis hypothesis to find to find directiondirection
.5000.5000-- .4960.4960
.0040.0040
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test of Mean (of Mean ( Unknown)Unknown)
8 8 -- 115115
of Mean (of Mean ( Unknown)Unknown)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
One Population TestsOne Population Tests
OnePopulation
8 8 -- 116116
Z Test(1 & 2tail)
t Test(1 & 2tail)
Z Test(1 & 2tail)
Mean Proportion Variance
2 Test(1 & 2tail)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
t Test for Mean t Test for Mean (( Unknown)Unknown)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Only Slightly Skewed & If Not Normal, Only Slightly Skewed & Large Sample (Large Sample (nn 30) Taken30) Taken
8 8 -- 117117
Large Sample (Large Sample (nn 30) Taken30) Taken2.2. Parametric Test ProcedureParametric Test Procedure
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
t Test for Mean t Test for Mean (( Unknown)Unknown)
1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed
If Not Normal, Only Slightly Skewed & If Not Normal, Only Slightly Skewed & Large Sample (Large Sample (nn 30) Taken30) Taken
8 8 -- 118118
Large Sample (Large Sample (nn 30) Taken30) Taken2.2. Parametric Test ProcedureParametric Test Procedure
3.3. t Test Statistict Test Statistic
tX
Sn
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
8 8 -- 119119
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
Given: n = 3; Given: n = 3; = .10= .10
8 8 -- 120120
t0
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
Given: n = 3; Given: n = 3; = .10= .10
8 8 -- 121121
t0
/2 = .05/2 = .05
/2 = .05/2 = .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
Given: n = 3; Given: n = 3; = .10= .10
df = n df = n -- 1 = 21 = 2
8 8 -- 122122
t0
/2 = .05/2 = .05
/2 = .05/2 = .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
v t.10 t.05 t.025
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
Critical Values of t Table Critical Values of t Table (Portion)(Portion)
Given: n = 3; Given: n = 3; = .10= .10
df = n df = n -- 1 = 1 = 22
8 8 -- 123123
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182t0
/2 = /2 = .05.05
/2 = .05/2 = .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
v t.10 t.05 t.025
TwoTwo--Tailed t TestTailed t TestFinding Critical t ValuesFinding Critical t Values
Critical Values of t Table Critical Values of t Table (Portion)(Portion)
Given: n = 3; Given: n = 3; = .10= .10
df = n df = n -- 1 = 21 = 2
8 8 -- 124124
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182t0 2.920-2.920
/2 = .05/2 = .05
/2 = .05/2 = .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestExample Example
Does an average box of Does an average box of cereal contain cereal contain 368368grams of cereal? A grams of cereal? A random sample of random sample of 3636
8 8 -- 125125
random sample of random sample of 3636boxes had a mean of boxes had a mean of 372.5372.5 & a standard & a standard deviation ofdeviation of 1212 grams. grams. Test at the Test at the .05.05 level.level. 368 gm.368 gm.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: :
HHaa: :
= = df = df =
Test Statistic: Test Statistic:
8 8 -- 126126
df = df = Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = df = df =
Test Statistic: Test Statistic:
8 8 -- 127127
df = df = Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = .05.05df = df = 36 36 -- 1 = 351 = 35
Test Statistic: Test Statistic:
8 8 -- 128128
df = df = 36 36 -- 1 = 351 = 35Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = .05.05df = df = 36 36 -- 1 = 351 = 35
Test Statistic: Test Statistic:
8 8 -- 129129
df = df = 36 36 -- 1 = 351 = 35Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.0301-2.0301
.025
Reject H0 Reject H0
.025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = .05.05df = df = 36 36 -- 1 = 351 = 35
Test Statistic: Test Statistic:
tX
Sn
372 5 3681236
2 25.
.
8 8 -- 130130
df = df = 36 36 -- 1 = 351 = 35Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.0301-2.0301
.025
Reject H0 Reject H0
.025
n 36
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = .05.05df = df = 36 36 -- 1 = 351 = 35
Test Statistic: Test Statistic:
tX
Sn
372 5 3681236
2 25.
.
8 8 -- 131131
df = df = 36 36 -- 1 = 351 = 35Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.0301-2.0301
.025
Reject H0 Reject H0
.025
n 36
Reject at Reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 368= 368HHaa: : 368368 = = .05.05df = df = 36 36 -- 1 = 351 = 35
Test Statistic: Test Statistic:
tX
Sn
372 5 3681236
2 25.
.
8 8 -- 132132
df = df = 36 36 -- 1 = 351 = 35Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.0301-2.0301
.025
Reject H0 Reject H0
.025
n 36
Reject at Reject at = .05= .05
There is evidence pop. There is evidence pop. average is not 368average is not 368
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t TestTailed t TestThinking ChallengeThinking Challenge
You work for the FTC. A You work for the FTC. A manufacturer of detergent manufacturer of detergent claims that the mean weight claims that the mean weight of detergent is of detergent is 3.253.25 lb. You lb. You
8 8 -- 133133
of detergent is of detergent is 3.253.25 lb. You lb. You take a random sample of take a random sample of 6464containers. You calculate the containers. You calculate the sample average to be sample average to be 3.2383.238lb. with a standard deviation lb. with a standard deviation of of .117.117 lb. At the lb. At the .01.01 level, is level, is the manufacturer correct?the manufacturer correct?
3.25 lb.3.25 lb.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: :
HHaa: :
df df
Test Statistic: Test Statistic:
8 8 -- 134134
df df Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 df df
Test Statistic: Test Statistic:
8 8 -- 135135
df df Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 .01.01df df 64 64 -- 1 = 631 = 63
Test Statistic: Test Statistic:
8 8 -- 136136
df df 64 64 -- 1 = 631 = 63Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 .01.01df df 64 64 -- 1 = 631 = 63
Test Statistic: Test Statistic:
8 8 -- 137137
df df 64 64 -- 1 = 631 = 63Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.6561-2.6561
.005
Reject H0 Reject H0
.005
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 .01.01df df 64 64 -- 1 = 631 = 63
Test Statistic: Test Statistic:
tX
Sn
3 238 3 2511764
82. .
..
8 8 -- 138138
df df 64 64 -- 1 = 631 = 63Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.6561-2.6561
.005
Reject H0 Reject H0
.005
n 64
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 .01.01df df 64 64 -- 1 = 631 = 63
Test Statistic: Test Statistic:
tX
Sn
3 238 3 2511764
82. .
..
8 8 -- 139139
df df 64 64 -- 1 = 631 = 63Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.6561-2.6561
.005
Reject H0 Reject H0
.005
n 64
Do not reject at Do not reject at = .01= .01
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
TwoTwo--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 3.25= 3.25HHaa: : 3.253.25 .01.01df df 64 64 -- 1 = 631 = 63
Test Statistic: Test Statistic:
tX
Sn
3 238 3 2511764
82. .
..
8 8 -- 140140
df df 64 64 -- 1 = 631 = 63Critical Value(s):Critical Value(s):
Decision:Decision:
Conclusion:Conclusion:
t0 2.6561-2.6561
.005
Reject H0 Reject H0
.005
n 64
Do not reject at Do not reject at = .01= .01
There is no evidence There is no evidence average is not 3.25average is not 3.25
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test of Mean (of Mean ( Unknown)Unknown)
8 8 -- 141141
of Mean (of Mean ( Unknown)Unknown)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t TestTailed t TestExample Example
Is the average capacity of Is the average capacity of batteries batteries at least 140 at least 140 ampereampere--hours? A random hours? A random sample of sample of 2020 batteries had batteries had
8 8 -- 142142
sample of sample of 2020 batteries had batteries had a mean of a mean of 138.47138.47 & a & a standard deviation of standard deviation of 2.662.66. . Assume a normal Assume a normal distribution. Test at the distribution. Test at the .05.05level.level.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: :
HHaa: :
==df =df =
Test Statistic: Test Statistic:
8 8 -- 143143
df =df =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = df = df =
Test Statistic: Test Statistic:
8 8 -- 144144
df = df =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = .05.05df = df = 20 20 -- 1 = 191 = 19
Test Statistic: Test Statistic:
8 8 -- 145145
df = df = 20 20 -- 1 = 191 = 19
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = .05.05df = df = 20 20 -- 1 = 191 = 19
Test Statistic: Test Statistic:
8 8 -- 146146
t0-1.7291
.05
Reject
df = df = 20 20 -- 1 = 191 = 19
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = .05.05df = df = 20 20 -- 1 = 191 = 19
Test Statistic: Test Statistic:
tX
Sn
138 47 1402 66
20
2 57.
..
8 8 -- 147147
t0-1.7291
.05
Reject
df = df = 20 20 -- 1 = 191 = 19
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 20
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = .05.05df = df = 20 20 -- 1 = 191 = 19
Test Statistic: Test Statistic:
tX
Sn
138 47 1402 66
20
2 57.
..
8 8 -- 148148
t0-1.7291
.05
Reject
df = df = 20 20 -- 1 = 191 = 19
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 20
Reject at Reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test SolutionSolution
HH00: : = 140= 140HHaa: : < 140< 140 = = .05.05df = df = 20 20 -- 1 = 191 = 19
Test Statistic: Test Statistic:
tX
Sn
138 47 1402 66
20
2 57.
..
8 8 -- 149149
t0-1.7291
.05
Reject
df = df = 20 20 -- 1 = 191 = 19
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 20
Reject at Reject at = .05= .05
There is evidence pop. There is evidence pop. average is less than 140average is less than 140
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t TestTailed t TestThinking ChallengeThinking Challenge
Youre a marketing analyst for Youre a marketing analyst for WalWal--Mart. WalMart. Wal--Mart had teddy Mart had teddy bears on sale last week. The bears on sale last week. The weekly sales ($ 00) of bears weekly sales ($ 00) of bears
8 8 -- 150150
weekly sales ($ 00) of bears weekly sales ($ 00) of bears sold in sold in 1010 stores was:stores was: 8 11 0 8 11 0 4 7 8 10 5 8 34 7 8 10 5 8 3. . At the At the .05.05 level, is there level, is there evidence that the average bear evidence that the average bear sales per store is sales per store is moremore thanthan 5 5 ($ 00)?($ 00)?
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: :
HHaa: :
= = df = df =
Test Statistic: Test Statistic:
8 8 -- 151151
df = df =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = df =df =
Test Statistic: Test Statistic:
8 8 -- 152152
df =df =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = .05.05df = df = 10 10 -- 1 = 91 = 9
Test Statistic: Test Statistic:
8 8 -- 153153
df = df = 10 10 -- 1 = 91 = 9
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = .05.05df = df = 10 10 -- 1 = 91 = 9
Test Statistic: Test Statistic:
8 8 -- 154154
t0 1.8331
.05
Reject
df = df = 10 10 -- 1 = 91 = 9
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = .05.05df = df = 10 10 -- 1 = 91 = 9
Test Statistic: Test Statistic:
tX
Sn
6 4 53 373
10
131..
.
8 8 -- 155155
t0 1.8331
.05
Reject
df = df = 10 10 -- 1 = 91 = 9
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 10
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = .05.05df = df = 10 10 -- 1 = 91 = 9
Test Statistic: Test Statistic:
tX
Sn
6 4 53 373
10
131..
.
8 8 -- 156156
t0 1.8331
.05
Reject
df = df = 10 10 -- 1 = 91 = 9
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 10
Do not reject at Do not reject at = .05= .05
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Tailed t Test Tailed t Test Solution*Solution*
HH00: : = 5= 5HHaa: : > 5> 5 = = .05.05df = df = 10 10 -- 1 = 91 = 9
Test Statistic: Test Statistic:
tX
Sn
6 4 53 373
10
131..
.
8 8 -- 157157
t0 1.8331
.05
Reject
df = df = 10 10 -- 1 = 91 = 9
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
n 10
Do not reject at Do not reject at = .05= .05
There is no evidence There is no evidence average is more than 5average is more than 5
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Z Test of ProportionZ Test of Proportion
8 8 -- 158158
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Data TypesData Types
Data
8 8 -- 159159
Numerical Qualitative
Discrete Continuous
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
1.1. Approximated by Approximated by Normal DistributionNormal Distribution
Excludes 0 or nExcludes 0 or n
Sampling Distribution Sampling Distribution of Proportionof Proportion
Sampling DistributionSampling Distribution
.2.2
.3.3P(PP(P^^ )) 13 ppnpn
8 8 -- 160160
pp
Excludes 0 or nExcludes 0 or n
2.2. MeanMean
3.3. Standard ErrorStandard Error
P p
where where pp00 = Population Proportion= Population Proportionpp^^pp
nn 11
.0.0
.1.1
.2.2
.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0
PP^^
00 00
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Standardizing Sampling Standardizing Sampling Distribution of ProportionDistribution of Proportion
Sampling Sampling DistributionDistribution
Standardized Standardized Normal DistributionNormal Distribution
ZZpp pp pp
pp pp
nn
^^pp
pp
^^
^^
(( ))11
^^00
00 00
8 8 -- 161161
Z Z = 0= 0
zz= 1= 1
ZZ
DistributionDistribution Normal DistributionNormal Distribution
PP^^PP
PP
^^
^^
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
One Population TestsOne Population Tests
OnePopulation
8 8 -- 162162
Z Test(1 & 2tail)
t Test(1 & 2tail)
Z Test(1 & 2tail)
Mean Proportion Variance
2 Test(1 & 2tail)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Sample Z Test Sample Z Test for Proportionfor Proportion
8 8 -- 163163
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Sample Z Test Sample Z Test for Proportionfor Proportion
1.1. AssumptionsAssumptions Two Categorical OutcomesTwo Categorical Outcomes
Population Follows Binomial DistributionPopulation Follows Binomial Distribution
8 8 -- 164164
Normal Approximation Can Be UsedNormal Approximation Can Be Used
Does Not Contain 0 or nDoes Not Contain 0 or n 13 ppnpn
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Sample Z Test Sample Z Test for Proportionfor Proportion
1.1. AssumptionsAssumptions Two Categorical OutcomesTwo Categorical Outcomes
Population Follows Binomial DistributionPopulation Follows Binomial Distribution
8 8 -- 165165
Normal Approximation Can Be UsedNormal Approximation Can Be Used
Does Not Contain 0 or nDoes Not Contain 0 or n
2.2. ZZ--test statistic for proportiontest statistic for proportion
Zp p
p pn
( )0
0 01Hypothesized Hypothesized population proportionpopulation proportion
13 ppnpn
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Example Example
The present packaging The present packaging system produces system produces 10%10%defective cereal boxes. defective cereal boxes. Using a new system, a Using a new system, a
8 8 -- 166166
Using a new system, a Using a new system, a random sample of random sample of 200200boxes hadboxes had1111 defects. defects. Does the new system Does the new system produce produce fewerfewer defects? defects? Test at the Test at the .05.05 level.level.
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: :
HHaa: :
= = nn ==
Test Statistic: Test Statistic:
8 8 -- 167167
nn ==
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp = .10= .10
HHaa: : pp < .10< .10
= = nn = =
Test Statistic: Test Statistic:
8 8 -- 168168
nn = =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp ==.10.10HHaa: : pp < .10< .10
= = .05.05nn = = 200200
Test Statistic: Test Statistic:
8 8 -- 169169
nn = = 200200
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp = .10= .10
HHaa: : pp < .10< .10
= = .05.05nn = = 200200
Test Statistic: Test Statistic:
8 8 -- 170170
nn = = 200200
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0-1.645
.05
Reject
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp = .10= .10
HHaa: : pp < .10< .10
= = .05.05nn = = 200200
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
11200
10
10 1 10212
8 8 -- 171171
nn = = 200200
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0-1.645
.05
Reject
n0 0
200
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp = .10= .10
HHaa: : pp < .10< .10
= = .05.05nn = = 200200
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
11200
10
10 1 10212
8 8 -- 172172
nn = = 200200
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0-1.645
.05
Reject Reject at Reject at = .05= .05
n0 0
200
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test SolutionSolution
HH00: : pp = .10= .10
HHaa: : pp < .10< .10
= = .05.05nn = = 200200
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
11200
10
10 1 10212
8 8 -- 173173
nn = = 200200
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0-1.645
.05
Reject Reject at Reject at = .05= .05
There is evidence new There is evidence new system < 10% defectivesystem < 10% defective
n0 0
200
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Thinking ChallengeThinking Challenge
Youre an accounting Youre an accounting manager. A yearmanager. A year--end audit end audit showed showed 4%4% of transactions of transactions had errors. You implement had errors. You implement
8 8 -- 174174
had errors. You implement had errors. You implement new procedures. A random new procedures. A random sample of sample of 500500 transactions transactions had had 2525 errors. Has the errors. Has the proportionproportion of incorrect of incorrect transactions transactions changedchanged at the at the .05.05 levellevel? ?
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: :
HHaa: :
= = nn = =
Test Statistic: Test Statistic:
8 8 -- 175175
nn = =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = nn = =
Test Statistic: Test Statistic:
8 8 -- 176176
nn = =
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = .05.05nn = = 500500
Test Statistic: Test Statistic:
8 8 -- 177177
nn = = 500500
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = .05.05nn = = 500500
Test Statistic: Test Statistic:
8 8 -- 178178
nn = = 500500
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = .05.05nn = = 500500
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
25500
04
04 1 04114
8 8 -- 179179
nn = = 500500
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
n0 0
500
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = .05.05nn = = 500500
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
25500
04
04 1 04114
8 8 -- 180180
nn = = 500500
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
Do not reject at Do not reject at = .05= .05
n0 0
500
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
OneOne--Proportion Z Test Proportion Z Test Solution*Solution*
HH00: : pp = .04= .04
HHaa: : pp .04.04 = = .05.05nn = = 500500
Test Statistic: Test Statistic:
Zp p
p p
( )
.
. ( . ).0
0 01
25500
04
04 1 04114
8 8 -- 181181
nn = = 500500
Critical Value(s):Critical Value(s):Decision:Decision:
Conclusion:Conclusion:
Z0 1.96-1.96
.025
Reject H0 Reject H0
.025
Do not reject at Do not reject at = .05= .05
There is no evidence There is no evidence proportion has proportion has changed from 4% changed from 4%
n0 0
500
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
One Population TestsOne Population Tests
OnePopulation
8 8 -- 182182
Z Test(1 & 2tail)
t Test(1 & 2tail)
Z Test(1 & 2tail)
Mean Proportion Variance
2 Test(1 & 2tail)
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Confidence Intervals, HypothesisConfidence Intervals, HypothesisTests, and pTests, and p--valuesvalues
All Start with Known Sampling Distribution forAll Start with Known Sampling Distribution forConfidence IntervalConfidence Interval
Pr( > given distance from ) = Pr( > given distance from ) = Draw an interval of size around actualDraw an interval of size around actual
X
X 2/zX
2/z
8 8 -- 183183
Draw an interval of size around actualDraw an interval of size around actual 11-- is the confidence levelis the confidence level
PP--ValueValue Assume true mean Assume true mean Pr( > measured distance) = pPr( > measured distance) = p
For oneFor one--sided value, no absolute valuesided value, no absolute value
Hypothesis testHypothesis test Pick , If p < , reject the null hypothesisPick , If p < , reject the null hypothesis
X
X
2/z
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Calculating Type II Error Calculating Type II Error ProbabilitiesProbabilities
8 8 -- 184184
ProbabilitiesProbabilities
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Power of TestPower of Test
1.1. Probability of Rejecting False HProbability of Rejecting False H00 Correct DecisionCorrect Decision
2.2. Designated 1 Designated 1 --
8 8 -- 185185
2.2. Designated 1 Designated 1 -- 3.3. Used in Determining Test AdequacyUsed in Determining Test Adequacy
4.4. Affected byAffected by True Value of Population ParameterTrue Value of Population Parameter
Significance Level Significance Level Standard Deviation & Sample Size Standard Deviation & Sample Size nn
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 1Step 1
Hypothesis:Hypothesis:HH00: : 00 368368HH11: : 00 < 368< 368 = .05= .05
n =n =15/15/2525
DrawDraw
8 8 -- 186186
XX00 = 368= 368
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerSteps 2 & 3Steps 2 & 3
Hypothesis:Hypothesis:HH00: : 00 368368HH11: : 00 < 368< 368 = .05= .05
n =n =15/15/2525
DrawDraw
8 8 -- 187187 XX11 = 360= 360
XX00 = 368= 368True Situation:True Situation:11 = 360= 360
DrawDraw
SpecifySpecify
11--
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 4Step 4
Hypothesis:Hypothesis:HH00: : 00 368368HH11: : 00 < 368< 368 = .05= .05
n =n =15/15/2525
DrawDraw
8 8 -- 188188 XX11 = 360= 360 363.065363.065
XX00 = 368= 368True Situation:True Situation:11 = 360= 360
065.363
25
1564.13680
n
ZXL
DrawDraw
SpecifySpecify
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 5Step 5
Hypothesis:Hypothesis:HH00: : 00 368368HH11: : 00 < 368< 368 = .05= .05
n =n =15/15/2525
DrawDraw
8 8 -- 189189 XX11 = 360= 360 363.065363.065
XX00 = 368= 368True Situation:True Situation:11 = 360= 360 = .154= .154
11-- =.846=.846
DrawDraw
SpecifySpecify
Z TableZ Table
065.363
25
1564.13680
n
ZXL
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
Power CurvesPower Curves
PowerPower PowerPowerHH00: : 00 HH00: : 00
8 8 -- 190190
PowerPower
Possible True Values for Possible True Values for 11 Possible True Values for Possible True Values for 11
Possible True Values for Possible True Values for 11
HH00: : ==00
= 368 in = 368 in ExampleExample
2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
ConclusionConclusion
1.1. Distinguished Types of Hypotheses Distinguished Types of Hypotheses
2.2. Described Hypothesis Testing ProcessDescribed Hypothesis Testing Process
3.3. Explained pExplained p--Value ConceptValue Concept
8 8 -- 191191
3.3. Explained pExplained p--Value ConceptValue Concept
4.4. Solved Hypothesis Testing Problems Solved Hypothesis Testing Problems Based on a Single SampleBased on a Single Sample
5.5. Explained Power of a TestExplained Power of a Test
End of Chapter
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