Hypothesis Testing

1
Hypothesis Testing

• Variance known?

2
Sampling Distribution

• Over-the-counter stock selling prices
• calculate average price of all stocks listed ?
• take a sample of 25 stocks and record price
• calculate average price of the 25 stocks x-bar
• take all possible samples of size 25
• would all x-bars be equal?
• average all the possible x-bars equals ?

3
Sampling Distribution
20
H0
4
Sampling Distribution
It is unlikely that we would get a sample mean of
this value ...
20
H0
5
Sampling Distribution
It is unlikely that we would get a sample mean of
this value ...
... if in fact this were the population mean
20
H0
6
Sampling Distribution
It is unlikely that we would get a sample mean of
this value ...
... therefore, we reject the hypothesis that ?
50.
... if in fact this were the population mean
20
H0
7
Null Hypothesis

• What is tested
• Always has equality sign ?, ???or ??
• Designated H0
• Example ... H0 ? ? 3

8
Alternative Hypothesis

• Opposite of null hypothesis
• Always has inequality sign ?,??, or ?
• Designated H1
• Example
• H1 ? lt 3

9
Decision

• Reject null hypothesis
• Retain, or, fail to reject, null hypothesis
• Do not use the term accept

10
p-value

• Probability of obtaining a test statistic more
  extreme (??or ???than actual sample value given
  H0 is true
• Called observed level of significance
• Smallest value of ? H0 can be rejected
• Used to make rejection decision
• If p-value ? ?, reject H0

11
Level of Significance

• Defines unlikely values of sample statistic if
  null hypothesis is true
• Called rejection region of sampling distribution
• Designated ??(alpha)
• Typical values are .01, .05, .10
• Selected by researcher at start

12
Rejection Region (one-tail test)
Sampling Distribution
Level of Confidence
1 - ?
13
Rejection Region (one-tail test)
Sampling Distribution
Level of Confidence
1 - ?
Observed sample statistic
14
Rejection Region (one-tail test)
Sampling Distribution
Level of Confidence
1 - ?
Observed sample statistic
15
Rejection Regions (two-tailed test)
Sampling Distribution
Level of Confidence
1 - ?
16
Rejection Regions (two-tailed test)
Sampling Distribution
Level of Confidence
1 - ?
Observed sample statistic
17
Rejection Regions (two-tailed test)
Sampling Distribution
Level of Confidence
1 - ?
Observed sample statistic
18
Rejection Regions (two-tailed test)
Sampling Distribution
Level of Confidence
1 - ?
Observed sample statistic
19
Risk of Errors in Making Decision

• Type I error
• Reject true null hypothesis
• Has serious consequences
• Probability of Type I error is alpha ?
• Called level of significance
• Type II error
• Do not reject false null hypothesis
• Probability of Type II error is beta ?

20
Decision Results
H0 Innocent
21
Hypothesis Testing

• State H0
• State H1
• Choose ?
• Choose n
• Choose test

22
Hypothesis Testing

• State H0
• State H1
• Choose ?
• Choose n
• Choose test

• Set up critical values
• Collect data
• Compute test statistic
• Make statistical decision
• Express decision

23
Two-tailed z-test

• Does an average box of cereal contain 368 grams
  of cereal? A random sample of 25 boxes has an
  average weight 372.5 grams. The company has
  specified ? to be 15 grams. Test at the .05
  level.

368 gm.
24
Two-tailed z-test
Test Statistic Decision Conclusion

• H0
• H1
• ? ?
• n ?
• Critical Value(s)

25
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ?
• n ?
• Critical Value(s)

26
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ? .05
• n ? 25
• Critical Value(s)

27
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ? .05
• n ? 25
• Critical Value(s)

28
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ? .05
• n ? 25
• Critical Value(s)

29
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ? .05
• n ? 25
• Critical Value(s)

Do not reject at ? .05
30
Two-tailed z-test
Test Statistic Decision Conclusion

• H0 ? 368
• H1 ? ? 368
• ? ? .05
• n ? 25
• Critical Value(s)

Do not reject at ? .05
No evidence average is not 368
31
Two-tailed z-test p-value
Z value of sample statistic
?
32
Two-tailed z-test p-value
p-value is P(z ? -1.50 or z ? 1.50)
Z value of sample statistic
?
33
Two-tailed z-test p-value
p-value is P(z ? -1.50 or z ? 1.50)
Z value of sample statistic
?
34
Two-tailed z-test p-value
p-value is P(z ? -1.50 or z ? 1.50)
.4332
Z value of sample statistic
From Z table lookup 1.50
?
?
35
Two-tailed z-test p-value
p-value is P(z ? -1.50 or z ? 1.50)
?
.5000- .4332 .0668
.4332
Z value of sample statistic
From Z table lookup 1.50
?
?
36
Two-tailed z-test p-value
p-value is P(z ? -1.50 or z ? 1.50) .1336
?
.5000- .4332 .0668
.4332
Z value of sample statistic
From Z table lookup 1.50
?
?
37
Two-tailed z-test p-value
1/2 p-value .0668
1/2 p-value .0668
1/2 ? .025
1/2 ? .025
38
Two-tailed z-test p-value
(p-Value .1336) ?? (? .05)
Do not reject.
1/2 p-Value .0668
1/2 p-Value .0668
1/2 ? .025
1/2 ? .025
Test statistic is in Do not reject region
39
Two-tailed z-test (? known) challenge

• You are a Q/C inspector. You want to find out if
  a new machine is making electrical cords to
  customer specification average breaking strength
  of 70 lb. with ? 3.5 lb. You take a sample of
  36 cords compute a sample mean of 69.7 lb. At
  the .05 level, is there evidence that the machine
  is not meeting the average breaking strength?

40
solution template (? known)
Test Statistic Decision Conclusion

• H0
• H1
• ?
• n
• Critical Value(s)

41
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0
• H1
• ?
• n
• Critical Value(s)

42
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ?
• n
• Critical Value(s)

43
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ? .05
• n 36
• Critical Value(s)

44
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ? .05
• n 36
• Critical Value(s)

45
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ? .05
• n 36
• Critical Value(s)

46
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ? .05
• n 36
• Critical Value(s)

Do not reject at ? .05
47
Two-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? 70
• H1 ? ? 70
• ? .05
• n 36
• Critical Value(s)

Do not reject at ? .05
No evidence average is not 70
48
One-tailed z-test (? known)

• Assumptions
• Population is normally distributed
• If not normal, can be approximated by normal
  distribution for large samples

49
One-tailed z-test (? known)

• Assumptions
• Population is normally distributed
• If not normal, can be approximated by normal
  distribution for large samples
• Null hypothesis has ? or ? sign only

50
One-tailed z-test (? known)

• Assumptions
• Population is normally distributed
• If not normal, can be approximated by normal
  distribution for large samples
• Null hypothesis has ? or ? sign only
• Z-test statistic

51
One-tailed z-test (? known)
H0?????0 H1 ??lt 0
Must be significantly below ?
52
One-tailed z-test (? known)
H0?????0 H1 ??lt 0
H0?????0 H1 ??gt 0
Must be significantly below ?
Small values satisfy H0 . Do not reject!
53
One-tailed z-test (? known)
What is z given ? .025?
? .025
?
54
One-tailed z-test (? known)
What Is Z given ? .025?
?
.500 - .025 .475
? .025
?
55
One-tailed z-test (? known)
Standardized Normal Probability Table (Portion)
What is z given ? .025?
?
?
.500 - .025 .475
.06
? .025
?
.4750
1.9
56
One-tailed z-test (? known)
Standardized Normal Probability Table (Portion)
What Is Z given ? .025?
?
?
.500 - .025 .475
.06
? .025
?
?
.4750
1.9
57
One-tailed z-test (? known)

• Does an average box of cereal contain more than
  368 grams of cereal? A random sample of 25 boxes
  showed?X 372.5. The company has specified ? to
  be 15 grams. Test at the .05 level.

368 gm.
58
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0
• H1
• ?
• n
• Critical Value(s)

59
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ?
• n
• Critical Value(s)

60
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ? .05
• n 25
• Critical Value(s)

61
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ? .05
• n 25
• Critical Value(s)

62
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ? .05
• n 25
• Critical Value(s)

63
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ? .05
• n 25
• Critical Value(s)

Do not reject at ? .05
64
One-tailed z-test (? known)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? gt 368
• ? .05
• n 25
• Critical Value(s)

Do not reject at ? .05
No evidence average is more than 368
65
One-tailed z-test (? known) p-value Solution
?
Use alternative hypothesis to find direction
?
Z value of sample statistic
66
One-tailed z-test (? known) p-value
?
Use alternative hypothesis to find direction
?
Z value of sample statistic
67
One-tailed z-test (? known) p-value
?
Use alternative hypothesis to find direction
.4332
?
?
Z value of sample statistic
From Z table lookup 1.50
68
One-tailed z-test (? known) p-value
?
?
Use alternative hypothesis to find direction
.5000- .4332 .0668
.4332
?
?
Z value of sample statistic
From Z table lookup 1.50
69
One-tailed z-test (? known) p-value
p-value .0668
?
?
Use alternative hypothesis to find direction
.5000- .4332 .0668
.4332
?
?
Z value of sample statistic
From Z table lookup 1.50
70
One-tailed z-test (? known) p-value
p-value .0668
? .05
71
One-tailed z-test (? known) p-value
(p-value .0668) ? (? .05).
Do not reject.
p-Value .0668
? .05
Test statistic is in Fail to reject region
72
p-value Challenge

• Youre an analyst for Ford. You want to find out
  if the average miles per gallon of Escorts is at
  least 32 mpg. Similar models have a standard
  deviation of 3.8 mpg. You take a sample of 60
  Escorts compute a sample mean of 30.7 mpg.
  What is the value of the observed level of
  significance (p-Value)?

73
p-value
p-value .004 p-value lt ? (? .01)
Reject H0.
?
?
.5000- .4960 .0040
Use alternative hypothesis to find direction
.4960
Z value of sample statistic
From Z table lookup 2.645
?
?
74
p-value

• Probability of obtaining a test statistic more
  extreme (??or ???than actual sample value given
  H0 is true
• Called observed level of significance
• Smallest value of ? H0 can be rejected
• Used to make rejection decision
• If p-value ? ?, reject H0

75
One-tailed t-test (? unknown)

• Does an average box of cereal contain less than
  the 368 grams indicated on the package? A random
  sample of 25 boxes showed?X 363.5 and s15.
  Test at the .05 level.

368 gr.
76

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0
• H1
• ?
• n
• Critical Value(s)

77

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ?
• n
• Critical Value(s)

78

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ? .05
• n 25, d.f. 24
• Critical Value(s)

79

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ? .05
• n 25, d.f. 24
• Critical Value(s)

80

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ? .05
• n 25, d.f. 24
• Critical Value(s)

81

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ? .05
• n 25, d.f. 24
• Critical Value(s)

Do not reject at ? .05
82

One-tailed t-test (? unknown)
Test Statistic Decision Conclusion

• H0 ? ? 368
• H1 ? lt 368
• ? .05
• n 25, d.f. 24
• Critical Value(s)

Do not reject at ? .05
No evidence average is less than 368
83
One-tailed t-test (? unknown) p-value Solution
?
Use alternative hypothesis to find direction
t value of sample statistic
?
84
One-tailed t-test (? unknown) p-value
From t table lookup -1.50 for 24 d.f.
?
?
Use alternative hypothesis to find direction
P-value 0.075
t value of sample statistic
?
85

One-tailed t-test (? unknown) p-value
p-value .075
? .05
86

One-tailed t-test (? unknown) p-value
(p-value .075) ? (? .05). Do
not reject.
p-value .075
Reject
? .05
Test statistic is in Fail to reject region
87
Questions?
88
ANOVA
89
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