1. Section 8.2
Day 3

2. Significance Testing
In a significance test, when is it appropriate to accept the null hypothesis, Ho?

3. Significance Testing
In a significance test, when is it appropriate to accept the null hypothesis, Ho?
You never accept the null hypothesis.
You either reject or do not reject the null hypothesis based on the evidence.

4. Components of a Significance Test for a Proportion
1. Give the name of the test and check the conditions for its use.
Name: Two-sided (or one-sided) significance test for a proportion
Sample is a simple random sample from a binomial population
Both npo and n(1 – po) are at least 10
Population size at least 10 times sample size

5. Components of a Significance Test for a Proportion
2. State the hypotheses, defining any symbols.
When testing a proportion, generically the null hypothesis, Ho, is:
Ho: the proportion of success, p, in the population from which the sample came is equal to po
Ho: p = po

6. 2. State Hypotheses
Alternative hypothesis, Ha, can be of 3 forms.

7. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is not equal to po
Ha: p po

8. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is greater than po
Ha: p > po

9. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is less than po
Ha: p < po

10. Components of Significance Test
3. Compute the test statistic, z, find the critical values, z*, and the P-value.
Include a sketch that illustrates the situation.

11. Components of Significance Test
3. Compute the test statistic, z, find the critical values, z*, and the P-value.
Include a sketch that illustrates the situation.

12. Components of Significance Test
4. Write a conclusion. Two parts:

13. Components of Significance Test
4. Write a conclusion. Two parts:
Compare value of z to predetermined critical values (z*) or compare P-value to α. Then say whether you reject null hypothesis or do not reject it, linking your reason to the critical values or P-value.

14. Components of Significance Test
4. Write a conclusion. Two parts:
Compare value of z to predetermined critical values (z*), or compare P-value to α. Then say whether you reject null hypothesis or do not reject it, linking your reason to the critical values or P-value.
Tell what your conclusion means in context of the situation.

15. Significance Test for Proportion
You want to test whether flipping a penny results in heads 50% of the time.
Suppose you flip a penny 40 times and get 17 heads.
What is the hypothesized standard, po?

16. Significance Test for Proportion
You want to test whether flipping a penny results in heads 50% of the time.
Suppose you flip a penny 40 times and get 17 heads.
What is the hypothesized standard, po?
po = 0.5

17. Significance Test for Proportion
You want to test whether flipping a penny results in heads 50% of the time.
Suppose you flip a penny 40 times and get 17 heads.
Name the test and check the conditions.

18. Significance Test for Proportion
You want to test whether flipping a penny results in heads 50% of the time.
Suppose you flip a penny 40 times and get 17 heads.
Name: Two-sided significance test for a proportion.
1st condition?

19. Random Sample?
We can assume the unbiased procedure of flipping a penny is random selection from a binomial population.
Suppose you flip a penny 40 times and get 17 heads.
2nd condition?

20. Both at least 10? npo = 40(0.5) = 20 n(1 – po) = 40(1 – 0.5) = 20
Both are at least 10
3rd condition?

21. Population Size? Is population at least 10 times the sample size?
Suppose you flip a penny 40 times and get 17 heads.

22. Population Size? Is population at least 10 times the sample size?
Yes. The population for flipping pennies can be assumed to be very, very large, certainly more than 10(40) = 400 flips.

23. State Hypotheses
Ho: p = 0.5, where p is the population proportion for getting heads when a coin is flipped

24. State Hypotheses
Ho: p = 0.5, where p is the population proportion for getting heads when a coin is flipped
Ha:

25. State Hypotheses
Ho: p = 0.5, where p is the population proportion for getting heads when a coin is flipped
Ha: p ≠ 0.5

26. Flipping a Penny Compute the test statistic and P-value.
Suppose you flip a penny 40 times and get 17 heads.

27. Test Statistic

28. Test Statistic

29. P-Value
Two-sided significance test:
P-value = =

30. P-Value Two-sided significance test: P-value = 2[normalcdf(-1EE99,-z)]=

31. P-Value Two-sided significance test:
P-value = 2[normalcdf(-1EE99, -z)]
= 2[normalcdf(-1EE99, )]
= 2( ) =

32. Use Calculator STAT TESTS 5: 1-PropZTest
For significance test of a proportion, we use the function 1-PropZTest
STAT
TESTS
5: 1-PropZTest

33. Use Calculator 1-PropZTest po: hypothesized value of p
x: number of successes
n: sample size
prop po < po > po choose one Ha
Calculate Draw

34. P-value Using Calculator
Get 17 heads out of 40, assume po = 0.5

35. P-value Using Calculator

36. Conclusion in Context Reject or do not reject Ho? P-value of 0.3428
z =

37. Conclusion
I do not reject the null hypothesis because the P-value of is greater than the significance level of α = 0.05.

38. Conclusion
I do not reject the null hypothesis because the P-value of is greater than the significance level of α = 0.05.
There is not sufficient evidence that the population proportion for getting heads when a coin is flipped is not equal to 0.5.

39. Conclusion
I do not reject the null hypothesis because the P-value of is greater than the significance level of α = 0.05.
There is not sufficient evidence that the population proportion for getting heads when a coin is flipped is not equal to 0.5.
Note: not sufficient evidence to support the alternate hypothesis.

40. Conclusion (Other Method)
I do not reject the null hypothesis at the 95% confidence level because the test statistic, z = is within the critical values interval of ± 1.96.
There is not sufficient evidence that the population proportion for getting heads when a coin is flipped differs from 0.5.

41. Page 514, E31

42. Page 514, E31
Name of test: Two-sided significance test for a proportion
“determine if there is convincing evidence that the true proportion of Americans who would be willing to pay $500 more for this purpose differs from two-thirds.”

43. Page 514, E31
Check conditions:
(1)

44. Page 514, E31 Check conditions: (1) The Gallup poll uses what you
can consider a simple random sample of
adults in the United States.

45. Page 514, E31 Check conditions: (1) The Gallup poll uses what you
can consider a simple random sample of
adults in the United States.
(2) Both np0 = 506 (2/3) and
n(1 – po) = 506 (1/3) are at least 10.

46. Page 514, E31 Check conditions: (1) The Gallup poll uses what you
can consider a simple random sample of
adults in the United States.
(2) Both np0 = 506 (2/3) and
n(1 – po) = 506 (1/3) are at least 10.
(3) There are more than 10(506) = 5060 adults in
the U.S. so population is 10 times sample size

47. Page 514, E31
State hypotheses. Make certain to define symbols.

48. Page 514, E31
State hypotheses. Make certain to define symbols.

49. Page 514, E31
State hypotheses. Make certain to define symbols.

50. Page 514, E31
Compute test statistic and draw a sketch.

51. Page 514, E31 Compute test statistic and draw a sketch.
P-value =

52. Page 514, E31 If use calculator: 1-PropZTest po: 2/3 x: 319 n: 506
prop po
Calculate
z = ; P-value = 0.084

53. Page 514, E31
Write conclusion in context.

54. Page 514, E31 Write conclusion in context.
I do not reject the null hypothesis because the P-value of 0.08 is greater than the significance level of α = 0.05.

55. Page 514, E31 There is not sufficient evidence that if all
Write conclusion in context.
I do not reject the null hypothesis because the P-value of 0.08 is greater than the significance level of α = 0.05.
There is not sufficient evidence that if all
U.S. adults had been asked, the proportion
who would have said they were willing to
pay $500 more per year would be different
from 2/3.

56. 2nd Method
I do not reject the null hypothesis at the 95% confidence level because the test statistic of z = is inside the critical values of z* =
There is not sufficient evidence that if all
U.S. adults had been asked, the proportion
who would have said they were willing to
pay $500 more per year would be different
from 2/3.

57. Page 511, P31

58. Page 511, P31 (a) normalcdf(-1EE99, -2.576) = .005
Two-sided test so must double the significance level.
.005(2) = .010 = 1%

59. Page 511, P31
(b) Significance level of 2% means 1% below and 1% above reasonably likely area.
invNorm(.01) = -2.33
Critical values are

60. Page 513, P42

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

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

63. 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.

64. 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

65. 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

66. 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

67. 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

68. 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.

69. 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
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.

70. Page 513, P45

71. Page 513, P45
(a) Always look at the data before writing the null and alternative hypotheses.

72. Page 513, P45
(a) Always look at the data before writing the null and alternative hypotheses.
False. This is considered cheating because,
for example, after looking at the data you could
realize that if you switched to a one-sided test,
you would get a smaller P-value and then could
reject H0.

73. Page 513, P45
(b) All else being equal, using a one-sided test will result in a larger P-value than using a two-sided test.

74. Page 513, P45
(b) All else being equal, using a one-sided test will result in a larger P-value than using a two-sided test.
False. The P-value for a one-sided test is
half that for the two-sided test.

75. Page 513, P45
(c) The P-value is the probability that the null hypothesis is true.

76. Page 513, P45
(c) The P-value is the probability that the null hypothesis is true.
False. Either the null hypothesis is true or it isn’t.
The P-value is the probability of getting a
sample statistic as extreme as or even more
extreme than the one you got from your
sample, assuming null hypothesis is true.

77. Page 513, P45
(d) A statistically significant result means the P-value is “small”.
True.

78. Page 511, P28

79. 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.

80. Page 511, P30

81. Page 511, P30
(a) Critical value associated with significance level of 0.12?
z* = invNorm(0.06) =

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

83. Page 511, P33

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

85. Page 511, P33
Step 2:

86. Page 511, P33
Step 3:

87. Page 511, P33
Step 3 (con’t):

88. 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 α = 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.

89. Page 512, P35

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

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

92. 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.

93. Page 515, E35

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

95. Page 515, E35

96. Page 515, E36

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

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

99. Page 516, E41

100. Page 516, E41

101. Page 516, E41

102. Page 516, E41

103. Page 516, E41

104. Questions?