1. Section 8.2
Day 4

2. Tomorrow: Quiz Use backside of note card used for 8.1 HW Quiz 8.1 – 8.2…when? Test 8.1 – 8.2…when? -- Both sides of 1 note card

3. Reject, not reject, or accept null hypothesis?
P-value = % confidence
2) P-value = % confidence
3) P-value = significance level of 0.1

4. Reject, not reject, or accept null hypothesis?
P-value = % confidence
P-value < α = 0.05 so reject Ho
2) P-value = % confidence
3) P-value = significance level of 0.1

5. Reject, not reject, or accept null hypothesis?
P-value = % confidence
P-value < α = 0.05 so reject Ho
2) P-value = % confidence
P-value > α = 0.01 so do not reject Ho
3) P-value = significance level of 0.1

6. Reject, not reject, or accept null hypothesis?
P-value = % confidence
P-value < α = 0.05 so reject Ho
2) P-value = % confidence
P-value > α = 0.01 so do not reject Ho
3) P-value = significance level of 0.1
P-value < α = 0.1 so reject Ho

7. Types of Errors
Reasoning of significance tests often compared to that of a jury trial.

8. Types of Errors
Reasoning of significance tests often compared to that of a jury trial.

9. Types of Errors

10. Types of Error and Power
Type I Error: rejecting a true null hypothesis

11. Types of Error and Power
Type I Error: rejecting a true null hypothesis
The probability of making a Type I error is equal to the significance level, α, of the test.
P(Type I error) = α

12. Types of Error and Power
Type I Error: To decrease the probability of a Type I error, make α smaller.
We make α smaller by increasing the confidence level.

13. Types of Error and Power
Type I Error: To decrease the probability of a Type I error, make α smaller.
We make α smaller by increasing the confidence level.
Confidence level α
90%
95%
99%

14. Types of Error and Power
Type I Error: To decrease the probability of a Type I error, make α smaller.
Changing sample size has no effect on the probability of a Type I error.

15. Types of Error and Power
If the null hypothesis is false, you can not make a Type I error.
Type I Error: rejecting a true null hypothesis

16. Types of Error and Power
Type II Error: Failing to reject a false null hypothesis.

17. Types of Error and Power
Type II Error: Failing to reject a false null hypothesis.
To decrease the probability of making Type II error:
Increase sample size or
Make significance level, α, larger

18. Types of Error and Power
If null hypothesis is true, you can not make a Type II error.

19. Types of Error and Power
Power: the probability of rejecting the null hypothesis.

20. Types of Error and Power
Power: the probability of rejecting the null hypothesis.
When null hypothesis is false, you want to reject it. Therefore you want the power to be large.

21. Types of Error and Power
When null hypothesis is false, you want to reject it. Therefore you want the power to be large.
Power = 1 – probability of Type II error
How do you increase the power?

22. Types of Error and Power
Power = 1 – probability of Type II error
How do you increase the power?
decrease probability of Type II error

23. Types of Error and Power
Power = 1 – probability of Type II error
How do you increase the power?
decrease probability of Type II error
Take larger sample or make α larger

24. Page 511, P30

25. Page 511, P30
critical value associated with significance level of 0.12?

26. Page 511, P30
(a) critical value associated with significance level of 0.12?
invNorm(0.06) is about ± 1.55

27. Page 511, P30
(b) What significance level is associated with critical values of z* of ?

28. Page 511, P30
(b) What significance level is associated with critical values of z* of ?
2[normalcdf(-1EE99, -1.73)] =

29. Page 513, P42

30. Page 513, P42 (a) Name: One-sided significance test for a proportion
One-sided because question asks whether poll results imply less than half of adults are satisfied.

31. Page 513, P42
Conditions:
Problem states this is a random sample

32. Page 513, P42 Conditions: Problem states this is a random sample
Both npo = 1000(0.5) = 500 and n(1 – po) = 1000(1 – 0.5) = 500 are at least 10.

33. Page 513, P42 Conditions: Problem states this is a random sample
Both npo = 1000(0.5) = 500 and n(1 – po) = 1000(1 – 0.5) = 500 are at least 10.
Number of adults in U.S. is at least 10(1000) = 10,000

34. Page 513, P42 (b) (Recall, null hypothesis has form p = po)
H0 : p = 0.5, where p is proportion of all
adults in the U.S. who would say they are
satisfied with the quality of K–12 education
in the nation

35. Page 513, P42 (b) (Recall, null hypothesis has form p = po)
H0 : p = 0.5, where p is proportion of all
adults in the U.S. who would say they are
satisfied with the quality of K–12 education
in the country
Ha: p < 0.5 Question asks whether poll results imply less than half of adults are satisfied.

36. Page 513, P42 (c) po = 0.5, x = 46%(1000) = 460, n = 1000, p < po
Use 1-PropZTest
z is approx ; P-value is approx

37. Page 513, P42 (d) No significance level was given, so use
α = 0.05 and confidence level of 95%.
I reject the null hypothesis because the
P-value of is less than the
significance level of α = 0.05.

38. Page 513, P42 (d) No significance level was given, so use
α = 0.05 and confidence level of 95%.
I reject the null hypothesis because the
P-value of is less than the
significance level of α = 0.05.
There is sufficient evidence to support the claim that less than a majority of adults in the United States are satisfied with the quality of K–12 education.

39. Page 513, P42 (d) I reject the null hypothesis because the
test statistic, z, of about is outside the
critical value of
There is sufficient evidence to support the claim that less than a majority of adults in the United States are satisfied with the quality of K–12 education.

40. Page 511, P33
Name of test?

41. Page 511, P33 Two-sided significance test for a proportion.
Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs.
Hypotheses in symbols?

42. Page 511, P33 Two-sided significance test for a proportion.
Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs.
Ho: p = 0.5

43. Page 511, P33 Two-sided significance test for a proportion.
Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs.
Ho: p = 0.5
Ha: p ≠ 0.5

44. Page 516, E41
Name of test?

45. Page 516, E41 One-sided significance test for a proportion.
Determine whether the increase in the proportion of mutations is statistically significant
Hypotheses in symbols?

46. Page 516, E41 One-sided significance test for a proportion.
Determine whether the increase in the proportion of mutations is statistically significant
Ho: p = 0.02

47. Page 516, E41 One-sided significance test for a proportion.
Determine whether the increase in the proportion of mutations is statistically significant
Ho: p = 0.02
Ha: p > 0.02

48. Page 514 E27
Name of test?

49. Page 514 E27 Two-sided significance test for a proportion
test the claim that 60% of the students in school carry backpacks to school

50. Page 515, E33

51. Page 515, E33

52. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis

53. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The hypothesis you assume is true in the test

54. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The hypothesis you assume is true in the test

55. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Rejecting Ho when Ho is really true

56. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Rejecting Ho when Ho is really true

57. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
z-score of the sample statistic

58. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
z-score of the sample statistic

59. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Half the width of the confidence interval

60. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Half the width of the confidence interval

61. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Not rejecting Ho when Ho is really false

62. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
Not rejecting Ho when Ho is really false

63. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The hypothesis you are seeking evidence for

64. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The hypothesis you are seeking evidence for

65. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The probability of seeing a result this extreme or more extreme

66. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
The probability of seeing a result this extreme or more extreme

67. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
statistic +/- (critical value of z)(std dev of statistic)

68. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
statistic +/- (critical value of z)(std dev of statistic)

69. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
A standard for Ho rejection: also called z*

70. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
A standard for Ho rejection: also called z*

71. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
A standard for Ho rejection: also called the alpha-level

72. Vocabulary A) margin of error B) z-statistic
C) Type I error D) Type II error
E) critical value of z F) significance level
G) P-value H) confidence interval
I) null hypothesis J) alternative hypothesis
A standard for Ho rejection: also called the alpha-level

73. Page 511, P33

74. Page 511, P33
Step 1: Two-sided significance test for a proportion.

75. Page 511, P33
Conditions:
(1) Problem states this was random sample

76. Page 511, P33 Conditions: (1) Problem states this was random sample
(2) npo = 500(0.5) = 250 and n(1 – po) = 500(0.5) = 250 are both at least 10

77. Page 511, P33 Conditions: (1) Problem states this was random sample
(2) npo = 500(0.5) = 250 and n(1 – po) = 500(0.5) = 250 are both at least 10
(3) Total number of bookstores in U.S. is greater than 10(500) = 5000.

78. Page 511, P33 Step 2: State hypotheses.
Ho: p = 0.5, where p is proportion of all U.S. bookstores that sell DVDs

79. Page 511, P33 Step 2: State hypotheses.
Ho: p = 0.5, where p is proportion of all U.S. bookstores that sell DVDs
Ha: p ≠ 0.5

80. Page 511, P33
Step 3: Compute test statistic and draw sketch

81. Page 511, P33 Step 3: Compute test statistic and draw sketch
1-PropZTest
po: 0.5
x: 265
n: 500
prop ≠ po

82. Page 511, P33 Step 3: Compute test statistic and draw sketch
1-PropZTest
po: 0.5
x: z ≈ 1.34
n: P-value ≈ 0.18
prop ≠ po

83. Page 511, P33
Not given level of significance so use  = 0.05 and critical values of z* = ± 1.96
Step 3 (con’t):

84. Page 511, P33
Step 4: Write conclusion in context.

85. Page 511, P33 Step 4: Write conclusion in context.
I do not reject the null hypothesis because the P-value of 0.18 is greater than the significance level of a = 0.05.

86. Page 511, P33 Step 4: Write conclusion in context.
I do not reject the null hypothesis because the P-value of 0.18 is greater than the significance level of a = 0.05.
There is not sufficient evidence to support the claim that the proportion of bookstores in the U.S. that sell DVDs is not half.

87. Page 513, P42
Hypotheses?

88. Page 513, P42
Ho: p = 0.5, where p is the proportion of adults in the U.S. who say they are satisfied with the quality of K-12 education in the nation.

89. Page 513, P42
Ho: p = 0.5, where p is the proportion of adults in the U.S. who say they are satisfied with the quality of K-12 education in the nation.
Ha: p < 0.5
Does this imply that less than a majority are satisfied?

90. Page 512, P35

91. Page 512, P35
B. The proportion of all households that are multigenerational this year.

92. Page 513, E25
Which of these is not a true statement?

93. Page 513, E25 Which of these is not a true statement?
(The 95% confidence interval is 0.82 to 0.96).
D. If 75% of all dogs wear a collar, then you are reasonably likely to get a result like the one from this sample.

94. Page 515, E35

95. Page 515, E35
Suppose the newspaper’s percentage is actually right.

96. Page 515, E35

97. Page 515, E36

98. Page 515, E36
Suppose the newspaper’s percentage is actually wrong.

99. Page 515, E36
Suppose the newspaper’s percentage is actually wrong.

100. Page 516, E41

101. Page 516, E41

102. Page 516, E41

103. Page 516, E41

104. Page 516, E41

105. Page 511, P28

106. Page 511, P28
B. Assuming that Ho is true, the P-value is the probability of observing a value of a test statistic at least as far out in the tails of the sampling distribution as is the value of the test statistic, z, from your sample.

107. Questions?

108. Suppose that in a random sample of 500 households, you find that 309 households have a computer.
Test the claim that 56.5% of all households in the United States have a computer.
Use a significance level of α = 0.01.

109. Name of Test
This is a two-sided significance test for a proportion.

110. Conditions
1) We are told this is a random sample from a binomial population.

111. Conditions
1) We are told this is a random sample from a binomial population.
2) npo = 500(0.565) = 282.5
n(1 – po) = 500(0.435) = 217.5; so both are at least 10

112. Conditions
1) We are told this is a random sample from a binomial population.
2) npo = 500(0.565) = 282.5
n(1 – po) = 500(0.435) = 217.5; so both are at least 10
3) the number of households in the U.S. is at least 10(500) = 5000

113. Hypotheses
Ho: p = 0.565, where p is the proportion of households in the U.S. that have a computer

114. Hypotheses
Ho: p = 0.565, where p is the proportion of households in the U.S. that have a computer
Ha: p

115. Suppose that in a random sample of 500 households, you find that 309 households have a computer.
Test the claim that 56.5% of all households in the United States have a computer.
Use a significance level of α = 0.01.

116. Test Statistic and P-value
1-PropZTest
po: 0.565
x: 309
n: 500
prop po

117. Test Statistic and P-value

118. Conclusion
Do not reject the null hypothesis at the 0.01 significance level because the P-value of is greater than 0.01.
There is not sufficient evidence to support the claim that the percent of all households in the U.S. that have computers is not 56.5%.

119. Conclusion
Do not reject the null hypothesis at the 99% confidence level because the test statistic of ± 2.39 is within the critical values interval of ±
There is not sufficient evidence to support the claim that the percent of all households in the U.S. that have computers is not 56.5%.