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Chi-Square ( 2 ) Test of Variance. Chi-Square ( 2 ) Test for Variance. 1.Tests One Population Variance or Standard Deviation 2.Assumes Population Is Approximately Normally Distributed 3.Null Hypothesis Is H 0 : 2 = 0 2. Chi-Square ( 2 ) Test for Variance. - PowerPoint PPT Presentation
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Chi-Square (Chi-Square (22) Test ) Test of Varianceof Variance
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Chi-Square (Chi-Square (22) Test) Testfor Variancefor Variance
1.1. Tests One Population Variance or Tests One Population Variance or Standard DeviationStandard Deviation
2.2. Assumes Population Is Approximately Assumes Population Is Approximately Normally DistributedNormally Distributed
3.3. Null Hypothesis Is HNull Hypothesis Is H00: : 22 = = 0022
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Chi-Square (Chi-Square (22) Test) Testfor Variancefor Variance
1.1. Tests One Population Variance or Tests One Population Variance or Standard DeviationStandard Deviation
2.2. Assumes Population Is Approximately Assumes Population Is Approximately Normally DistributedNormally Distributed
3.3. Null Hypothesis Is HNull Hypothesis Is H00: : 22 = = 0022
4.4. Test StatisticTest Statistic
Hypothesized Pop. VarianceHypothesized Pop. Variance
Sample VarianceSample Variance
22
22
22
1)1)
(n(n SS
00
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Chi-Square (Chi-Square (22) ) DistributionDistribution
Select simple randomsample, size n.
Compute s2
Compute 2 =(n-1)s 2/2
Astronomical numberof 2 values
PopulationSampling Distributionsfor Different SampleSizes
21 2 30
Select simple randomsample, size n.
Compute s2
Compute 2 =(n-1)s 2/2
Astronomical numberof 2 values
PopulationSampling Distributionsfor Different SampleSizes
21 2 30
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Finding Critical Finding Critical Value ExampleValue Example
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20 20
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20
Reject
20
Reject
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20 5.991
Reject
20 5.991
Reject
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
What is the critical What is the critical 22 value given: value given:HHaa: : 22 > 0.7 > 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Finding Critical Finding Critical Value ExampleValue Example
What Do You Do If the Rejection Region Is on the Left?
What Do You Do If the Rejection Region Is on the Left?
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
20 20
22 Table Table (Portion)(Portion)
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 20
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
RejectReject
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 20
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
RejectReject Upper Tail Area Upper Tail Area for Lower Critical for Lower Critical Value = 1-.05 = .95Value = 1-.05 = .95
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 20
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
RejectReject Upper Tail Area Upper Tail Area for Lower Critical for Lower Critical Value = 1-.05 = .95Value = 1-.05 = .95
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 20
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
RejectReject Upper Tail Area Upper Tail Area for Lower Critical for Lower Critical Value = 1-.05 = .95Value = 1-.05 = .95
dfdf = = nn - 1 = 2 - 1 = 2
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 20
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
RejectReject Upper Tail Area Upper Tail Area for Lower Critical for Lower Critical Value = 1-.05 = .95Value = 1-.05 = .95
dfdf = = nn - 1 = 2 - 1 = 2
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
20 .103 20 .103
What is the critical What is the critical 22 value given: value given:HHaa: : 22 << 0.7 0.7
nn = 3 = 3 =.05? =.05?
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Upper Tail AreaDF .995 … .95 … .051 ... … 0.004 … 3.8412 0.010 … 0.103 … 5.991
Finding Critical Finding Critical Value ExampleValue Example
= .05= .05
22 Table Table (Portion)(Portion)
dfdf = = nn - 1 = 2 - 1 = 2
Upper Tail Area Upper Tail Area for Lower Critical for Lower Critical Value = 1-.05 = .95Value = 1-.05 = .95
RejectReject
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Chi-Square (Chi-Square (22) Test ) Test Example Example
Is the variation in boxes Is the variation in boxes of cereal, measured by of cereal, measured by the the variancevariance, equal to , equal to 1515 grams? A random grams? A random sample of sample of 2525 boxes had boxes had a standard deviation ofa standard deviation of 17.717.7 grams. Test at the grams. Test at the .05.05 level. level.
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: :
HHaa: :
= =
df = df =
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= =
df = df =
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
/2 = .025/2 = .025
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200 39.36439.36412.40112.401
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
/2 = .025/2 = .025
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200 39.36439.36412.40112.401
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
/2 = .025/2 = .025
2222
22
22
22
1)1) (25 -(25 -1)1) 1717 77
1515
3333 4242
(n(n SS
00
..
..
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200 39.36439.36412.40112.401
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
Do Not Reject at Do Not Reject at = .05 = .05 /2 = .025/2 = .025
2222
22
22
22
1)1) (25 -(25 -1)1) 1717 77
1515
3333 4242
(n(n SS
00
..
..
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
2200 39.36439.36412.40112.401
Chi-Square (Chi-Square (22) Test ) Test SolutionSolution
HH00: : 22 = 15 = 15
HHaa: : 22 15 15
= = .05.05
df = df = 25 - 1 = 2425 - 1 = 24
Critical Value(s):Critical Value(s):
Test Statistic: Test Statistic:
Decision:Decision:
Conclusion:Conclusion:
Do Not Reject at Do Not Reject at = .05 = .05
There Is No Evidence There Is No Evidence 22 Is Not 15 Is Not 15
/2 = .025/2 = .025
2222
22
22
22
1)1) (25 -(25 -1)1) 1717 77
1515
3333 4242
(n(n SS
00
..
..
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Calculating Type II Calculating Type II Error ProbabilitiesError Probabilities
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© 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 -
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
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
XX00 = 368= 368
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 1Step 1
Hypothesis:Hypothesis:HH00: : 00 368 368
HH11: : 00 < 368 < 368 = .05= .05
n =n =15/15/2525
DrawDraw
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
XX11 = 360= 360
XX00 = 368= 368
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerSteps 2 & 3Steps 2 & 3
Hypothesis:Hypothesis:HH00: : 00 368 368
HH11: : 00 < 368 < 368
‘‘True’ Situation:True’ Situation: 11 = 360 = 360
= .05= .05
n =n =15/15/2525
DrawDraw
DrawDraw
SpecifySpecify
1-1-
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
XX11 = 360= 360 363.065363.065
XX00 = 368= 368
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 4Step 4
Hypothesis:Hypothesis:HH00: : 00 368 368
HH11: : 00 < 368 < 368
‘‘True’ Situation:True’ Situation: 11 = 360 = 360
065.363
25
1564.13680
n
ZX L
065.363
25
1564.13680
n
ZX L
= .05= .05
n =n =15/15/2525
DrawDraw
DrawDraw
SpecifySpecify
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
XX11 = 360= 360 363.065363.065
XX00 = 368= 368
RejectRejectDo NotDo NotRejectReject
Finding PowerFinding PowerStep 5Step 5
Hypothesis:Hypothesis:HH00: : 00 368 368
HH11: : 00 < 368 < 368
‘‘True’ Situation:True’ Situation: 11 = 360 = 360
= .05= .05
n =n =15/15/2525
= .154= .154
1-1- =.846 =.846
DrawDraw
DrawDraw
SpecifySpecify
Z TableZ Table
065.363
25
1564.13680
n
ZX L
065.363
25
1564.13680
n
ZX L
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© 2003 Pearson Prentice Hall© 2003 Pearson Prentice Hall
Power CurvesPower Curves
PowerPower PowerPower
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 HH00: : 00
HH00: : = =00
= 368 in = 368 in
ExampleExample
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© 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 p-Value ConceptExplained p-Value 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
Any blank slides that follow are blank intentionally.