Inferences Based on a Single Sample: Tests of Hypothesis Chapter 9

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????????????

• Inferences Based on a Single Sample Tests of
  HypothesisChapter 9

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????

• 1. ???????
• ????????
• 3. P?p-value??????
• ?????????????
• 5. ?????????

3
???????
4
???????Hypothesis Testing Concepts
5
????Hypothesis Testing
6
????Hypothesis Testing
Population
?
?
?
?
?
?
?
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????Hypothesis Testing
??????????? 50 (hypothesis).
Population
?
?
?
?
?
?
?
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????Hypothesis Testing
??????????? 50 (hypothesis).
Population
?
?
?
?
?
?
???? Random sample
?
Mean ?X 20
?
?
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????Hypothesis Testing
??????????? 50 (hypothesis).
??Reject hypothesis! ????
Population
?
?
?
?
?
?
???? Random sample
?
Mean ?X 20
?
?
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????Hypothesis?

• 1.???????????A belief about a population
  parameter
• ????????population mean, proportion, variance
• ????????Must be stated before analysis

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????Hypothesis?

• 1.???????????A belief about a population
  parameter
• ????????population mean, proportion, variance
• ????????Must be stated before analysis

?????????? GPA ? 3.5!
?1984-1994 T/Maker Co.
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????Null Hypothesis

• 1. ??????????,??????
• 2. ????????????????
• 3. ??????? ?, ?, or ??
• 4. ?????H0
• 5. ?? H0 ? ? ????
• ???????(????)
• ?H0 ? ? 3

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????Alternative Hypothesis

• 1. ???????
• 2. ??????(???)? ?,??, or ?
• 3. ????Ha (????H1)
• 4. ??Ha ? lt ????
• ?, Ha ? lt 3

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????????????Identifying Hypotheses Steps

• ??Steps
• 1.??????????
• 2.???????????????
• 3.???????????
• ?, lt, or gt
• 4.?????????????

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????????????Identifying Hypotheses Steps

• ??
• ????????????????????????2?,???????????????,??????
  ????
• Ha ? ? 2
• H0 ? 2

• ??Steps
• 1.??????????
• 2.???????????????
• 3.???????????
• ?, lt, or gt
• 4.?????????????

16
????????????Identifying Hypotheses Steps

• ??Steps
• 1.??????????
• 2???????????????
• 3.???????????
• ?, lt, or gt
• 4.?????????????

• ??
• ???????????????????????20??
• Ha ? lt 20
• H0 ? ? 20

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????????????Identifying Hypotheses Steps

• ??Steps
• 1.??????????
• 2.???????????????
• 3.???????????
• ?, lt, or gt
• 4.?????????????

• ??
• ??????????????,????????????????????????????24?,???
  ?????????????,?????????????????????????
• Ha ? ? 24
• H0 ? ? 24

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?????????????????????µ?????

• H0 ? ? ?0,Ha ? gt ?0
• ????(right-tailed test)
• H0 ? ? ?0,Ha ? lt ?0
• ????( left-tailed test )
• H0 ? ?0,Ha ? ? ?0
• ????( two-tailed test )
• ?0?????????
• ??1. 2.???????

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??????(relibility)???

• ????(Confidence Level )
• ????(Error Bound)

20
?????????

• ??????Decision Making Risks

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??????Errors in Making Decision

• 1. ?I??Type I error
• ??????????reject a true null hypothesis
• ?I???????(alpha)??
• ??????????Called level of significance
• 2. ?II??Type II error
• ??????????Do not reject false null hypothesis
• ?II???????(beta)??

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???(power of the test)

• ?????????????
• P(reject Ho I Ha is true)
• 1 - b

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??

• ???????????????????????20??
• H0 ? ? 20 Ha ? lt 20
• ?I??
• ??????????????????????20?
• ????????????????????20?
• ? II ??
• ????????????????????20?
• ??????????????????????20?

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??????Decision Results
H0 ??????Innocent
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? ? ?????????
26
? ? ?????????
?
?
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? ? ?????????
?
?
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? ? ?????????
?
?
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? ? ?????????
????????I??II??
?
?
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??????Decision Results
H0 ??????Innocent
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????Level of Significance

• 1. ??????? I ????
• 2. ?????(alpha)?????????
• 3. ???????????
• ????? .01, .05, .10

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???????
33
???????
One
Population
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???????
One
Population
Mean
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???????
One
Population
Mean
Proportion
36
???????
One
Population
Mean
Proportion
Large
sknown
Sample
Z Test
(1 2
tail)
37
???????
One
Population
Mean
Proportion
Small
sunknown
Large
sknown
Sample
Sample
Z Test
t Test
(1 2
(1 2
tail)
tail)
38
???????
One
Population
Mean
Proportion
Small
sunknown
Large
sknown
Sample
Sample
Z Test
t Test
(1 2
(1 2
tail)
tail)
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????
40
????
What is Z given ? .025?
? .025
?
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????
?? .025 ? Z ??
?
.500 - .025 .475
? .025
?
a
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????
?? .025 ? Z ??
??????? (??)
?
?
.500 - .025 .475
.06
? .025
?
0.025
1.9
.4750
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????
?? .025 ? Z ??
??????? (??)
?
?
.500 - .025 .475
.06
? .025
?

0.025
?
1.9
.4750
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???????????Hypothesis Testing Steps
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???????????

1. ?????? H0
2. ?????? H1
3. ?????? ?
4. ????? n
5. ?????????

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???????????

• ?????rejection region???? critical values
• ????????
• ???????????
• ?????????????

1. ?????? H0
2. ?????? H1
3. ?????? ?
4. ????? n
5. ?????????

47
s????????????One-Tailed Z Test of Mean (s
known)
48
???????

• ??????????????,?????Escorts??????????????????32
  mpg(?????)??????????????????3.8 mpg??????? 60
  ?Escorts????????????? 30.7 mpg? ??????a0.01
  ?????????????32 mpg?

49
???????????

1. ?????? H0
2. ?????? H1
3. ?????? ?
4. ????? n
5. ?????????

H0µ?32 H1µlt32 a 0.01 n 60
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??????????(Test Statistic)

• ???????????????????????,????????????
• ????µ?,x ??????????

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???????????

• ?????rejection region???? critical values
• ????????
• ???????????
• ?????????????

1. ?????? H0
2. ?????? H1
3. ?????? ?
4. ????? n
5. ?????????

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????????(rejection region)????(critical value)

• ??? X lt C
• C ?????
• ??????????????????????

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????X ?????

• ????????????????(n?30)
• X ??????(???)
• ???? N(µ,s/vn)

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?????Rejection Region (One-Tail Test)
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Ho
Sample Statistic
Value
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Rejection
Region
Ho
Sample Statistic
Value
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Rejection
Region
Nonrejection Region ???
Ho
Sample Statistic
Value
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Rejection
Region
Nonrejection
Region
Ho
Sample Statistic
Value
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Rejection
Region
?
Nonrejection
Region
Ho
Sample Statistic
Value
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
1 - ?
??????
??????????Observed sample statistic
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?????Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
1 - ?
???????
??????????Observed sample statistic
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??????
63
??????
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??????????????

• X 30.7 lt 30.86
• ?????
• ????????( H0µ?32 )

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???????????

• ?0.01??????,
• ????????
• Escorts????????????????32 mpg(?? / ??)?

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?????

• µgt32 ??? ???????
• 2. ???? ??????????

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?????

• µgt32 ??? ???????
• Ans µgt32 ??? ???????
• µ32 ??? ??????
• 2. ???? ??????????
• Ans ??,??? ????????

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????????

• ??? X ????????????
• Standardized Test Statistic
• Z (X -µ)/ (s/ vn)

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????????

• ??? X ????????????
• Test Statistis
• Z (X -µ)/ (s/ vn)
• Z?????????? N(0 ,1)

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????????

• ??? X ????????????
• Test Statistis
• Z (X -µ)/ (s/ vn)
• Z?????????? N(0 ,1)
• ????????Zlt- Za

71
???????

• ??????????????,?????Escorts??????????????????32
  mpg(?????)??????????????????3.8 mpg??????? 60
  ?Escorts????????????? 30.7 mpg? ??????a0.01
  ?????????????32 mpg?

72
??????????

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
73
??Z????

• H0 ? 32
• Ha ? lt 32
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
74
??Z????

• H0 ? 32
• Ha ? lt 32
• ? .01
• n 60
• Critical Value(s)

Test Statistic Decision Conclusion
75
??Z????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ? 0.10
• n 48
• Critical Value(s)

Test Statistic Decision Conclusion
-1.28
76
??Z????

• H0 ? 32
• Ha ? lt 32
• ? .01
• n 60
• Critical Value(s)

Test Statistic Decision Conclusion
77
??Z????

• H0 ? 32
• Ha ? lt 32
• ? .01
• n 60
• Critical Value(s)

Test Statistic Decision Conclusion
?? .01???????
78
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 25
• Critical Value(s)

Test Statistic Decision Conclusion
? ? .05?????Ho
???????????368??
79
s?????????????

• ???????????,???????60??????????,?????48??????????
  ?????,????????58,???????????11,??0.10???????????
  ???????????????

80
s??????????????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
81
s??????????????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ? 0.10
• n 48
• Critical Value(s)

Test Statistic Decision Conclusion
82
??Z????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ? 0.10
• n 48
• Critical Value(s)

Test Statistic Decision Conclusion
-1.28
83
??Z????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ? .10
• n 48
• Critical Value(s)

Test Statistic Decision
0.58-0.60
1.26
0.11
48
-1.28
84
??Z????

• H0 ? ? 0.60
• Ha ? lt 0.60
• ? .10
• n 48
• Critical Value(s)

Test Statistic Decision ?? 0 .10?????????
0.58-0.60
1.26
0.11
48
-1.28
85
??Z????

• Conclusion
• ?0.1??????,?????????
• ??????????????

86
??Z????

• ????????????Ho?
• ????
• ????????????
• ?????????????????
• ????????????????

87
??Z????

• ????????????Ho?
• ????
• ????????????
• ?????????????????
• ????????????????
• ??
• ?????????II????

88
??????

• ????????
• ???????????, b??
• ?I??, ????
• ?????b??? ?????b???
• 3. ??????, ?
• ? ????b???
• 4. ???, n
• ?n???b??

89
??????

• ????????????????????368 ????
• ?????36 ??????? ???????X 372.5???
  ????????????????15?????????a 0.05????

368 gm.
90
s??????????????

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
91
s??????????????

• H0 ? 368
• Ha ? gt 368
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
92
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 36
• Critical Value(s)

Test Statistic Decision Conclusion
93
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 36
• Critical Value(s)

Test Statistic Decision Conclusion
94
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 25
• Critical Value(s)

Test Statistic Decision Conclusion
95
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 25
• Critical Value(s)

Test Statistic Decision Conclusion
? ? .05?????Ho
96
s??????????????

• H0 ? 368
• Ha ? gt 368
• ? .05
• n 25
• Critical Value(s)

Test Statistic Decision Conclusion
? ? .05?????Ho
???????????368??
97
s????????????
H0?????µ0 Ha ??ltµ0
H0? ????µ0 Ha ?? gt µ0
??????????- Za???H0
??????????Za???H0
98
?? ??????

• ???????????????????????????25?
  ????,?????????????????3? / ??,?????30???????????
  ????????24,??????????
• a.???????????
• b.?????????????????????24.5?,?????II???????

99
??Z?? ?????? ??

• ???????????I????
• Hoµ?25 Haµlt25
• P(?I????)
• P(??HoHo??)
• P(X lt 24µ 25)
• P(Z lt -1.83) 0.0336

100
??Z?? ?????? ??

• b.

Hoµ?25 Haµlt25
P(?II????) P(??HoHa??)
P(X ? 24µ 24.5)
P(Z ?- 0.91) 0.6814