1. Significance Test for the Difference of Two Proportions
Section 8.4
Significance Test for the Difference of Two Proportions

2. Critical Values
For a 2-sided test, what are the critical values for a significance level of α = 0.05?

3. Critical Values
For a 2-sided test, what are the critical values for a significance level of α = 0.05?

4. Critical Values
For a 2-sided test, what are the critical values for a significance level of α = 0.05?

5. Critical Values
For a 2-sided test, what are the critical values for a significance level of α = 0.05?
invNorm (.025) = -1.96
invNorm (.975) = 1.96
z* = ± 1.96

6. Critical Values
For a one-sided test, what is the critical value for a significance level of α = 0.05?

7. Critical Values
For a one-sided test, what is the critical value for a significance level of α = 0.05?

8. Critical Values
For a one-sided test, what is the critical value for a significance level of α = 0.05?
If Ha: p < po, invNorm (.05 ) =

9. Critical Values
For a one-sided test, what is the critical value for a significance level of α = 0.05?

10. Critical Values
For a one-sided test, what is the critical value for a significance level of α = 0.05?
If Ha: p > po, invNorm (.95 ) =

11. Components of a Significance Test for the Difference of Two Proportions for Surveys
Four parts:

12. Components of a Significance Test for the Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions

13. Components of a Significance Test for the Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses

14. Components of a Significance Test for the Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses
3. Compute test statistic and P-value

15. Components of a Significance Test for the Difference of Two Proportions for Surveys
Four parts:
1. Name test and check conditions
2. Write null and alternative hypotheses
3. Compute test statistic and P-value
4. Write a conclusion in context

16. Name Test & Check Conditions
Name: Use the whole name!
One-sided significance test for the difference of two proportions or
Two-sided significance test for the difference of two proportions

17. Check Conditions First condition:
Random samples selected independently from two different populations

18. Check Conditions Second condition: normal?
All these quantities must be at least 5.
Show all 4 calculations and state significance of results

19. Check Conditions Third condition:
Each population is at least 10 times as large as its sample size.
Remember to explain why this is reasonable.

20. Null Hypothesis
Null hypothesis almost always one of “no difference” or “no effect”

21. Null Hypothesis
Null hypothesis almost always one of “no difference” or “no effect”
Several choices as long as define p1 and p2
Ho: The proportion of successes p1 in the first population is equal to the proportion of successes p2 in the second population.
Write Ho in context of situation.

22. Null Hypothesis
Null hypothesis almost always one of “no difference” or “no effect”
Ho: p1 = p2, where p1 is the proportion of successes in the first population and p2 is the proportion of successes in the second population.
Write Ho in context of situation.

23. Null Hypothesis
Null hypothesis almost always one of “no difference” or “no effect”
Ho: p1 - p2 = 0, where p1 is the proportion of successes in the first population and p2 is the proportion of successes in the second population.
Write Ho in context of situation.

24. Alternative Hypothesis
Form depends on whether you need a two-sided or one-sided test.

25. Alternative Hypothesis
Two-sided test
Ha: The proportion of successes p1 in the first population is not equal to the proportion of successes p2 in the second population.
In symbols:
Ha: p1 ≠ p2 or Ha: p1 – p2 ≠ 0

26. Alternative Hypothesis
One-sided test
Ha: The proportion of successes p1 in the first population is greater than the proportion of successes p2 in the second population.
In symbols:
Ha: p1 > p2 or Ha: p1 – p2 > 0

27. Alternative Hypothesis
One-sided test
Ha: The proportion of successes p1 in the first population is less than the proportion of successes p2 in the second population.
In symbols:
Ha: p1 < p2 or Ha: p1 – p2 < 0

28. Compute Test Statistic and P-value
If we assume p1 = p2, then use

29. Compute Test Statistic and P-value
If we assume p1 = p2, then use

30. Compute Test Statistic and P-value
If we assume p1 = p2, then use

31. is called “pooled estimate” of the common proportion of successes

32. Compute Test Statistic and P-value
or use 2-PropZTest
Only valid if we assume p1 = p2

33. Compute Test Statistic and P-value
If do not assume p1 = p2, then use:

34. Compute Test Statistic and P-value
The P-value is the probability of getting a value of z as extreme or more extreme than that from your samples if Ho is true.
Include a sketch.

35. Write a Conclusion in Context
State whether you reject or do not reject the null hypothesis.

36. Write a Conclusion in Context
State whether you reject or do not reject the null hypothesis.
Link this conclusion to your computations by
(1) comparing z to critical value z*
reject null hypothesis if z is more extreme than z* or

37. Write a Conclusion in Context
State whether you reject or do not reject the null hypothesis.
Link this conclusion to your computations by
(1) comparing z to critical value z*
reject null hypothesis if z is more extreme than z* or
(2) comparing P-value to level of significance α
reject null hypothesis if P-value is smaller than α

38. Write a Conclusion in Context
State whether you reject or do not reject the null hypothesis.
Link this conclusion to your computations by (1) comparing z to critical value z*, reject null hypothesis if z is more extreme than z* or
(2) comparing P-value to level of significance α, reject null hypothesis if P-value is smaller than α
Write sentence giving conclusion in context related to alternative hypothesis.

39. Page 536, P50

40. Page 536, P50
Give appropriate statistical evidence to support your answer means complete all 4 steps for a significance test.

41. Page 536, P50 Name: one-sided significance test for the
difference of two proportions
because you are asked whether the data
support the conclusion that there was a
decrease in voter support for the
candidate.

42. Page 536, P50 We are told that we have two random
samples from a population of probable
voters in some city.
It’s reasonable to assume that the samples
are independent because the random
samples were taken about 2 weeks apart.

43. Page 536, P50 Let n1 be 1st sample and n2 the 2nd sample.
Each of the following is at least 5:

44. Page 536, P50 The number of probable voters at both times
should be larger than 10 times the sample
size for both samples, and it is unknown
whether this is the case.

45. Page 536, P50
Using symbols, what is the null hypothesis?

46. Page 536, P50
Ho: p1 = p2

47. Page 536, P50 Ho: p1 = p2, where p1 is the proportion of all
probable voters who favored the candidate
at the time of the first survey and p2 is the
proportion of all probable voters who
favored the candidate one week before the
election.

48. Page 536, P50
Ha: p1 > p2
We want to see if there was a decrease in voter support from 1st survey to 2nd survey.

49. Test Statistic and P-value
2-PropZTest
x1: 321
n1: 600
x2: 382
n2: 750
p1 > p2
Calculate
Survey 3 weeks before start of campaign
2nd survey

50. Test Statistic and P-value
2-PropZTest
x1: 321
n1: 600
x2: 382
n2: 750
p1 > p2
Calculate
z = and P-value =
Survey 3 weeks before start of campaign
2nd survey

51. Page 536, P50

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

53. Write Conclusion in Context
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 to conclude that there was a decrease in voter support for the new candidate after the parking tickets were revealed.

54. For this one-sided test:
What critical value(s) of z would you use to compare against the test statistic of z = for a 95% confidence level?

55. Z*
What critical value(s) of z would you use to compare against the test statistic of z = for a 95% confidence level?
One-sided test at 95% confidence level:
z* = invNorm(0.95) = 1.64

56. Write Conclusion in Context
I do not reject the null hypothesis because the test statistic of z = is less extreme than the critical value of z* = 1.64.
There is not sufficient evidence to conclude that there was a decrease in voter support for the new candidate after the parking tickets were revealed.

57. Problem # 1 If the P-value of a test is less than the
level of significance, then which of
these conclusions is correct?

58. C. The null hypothesis is true.
If the P-value of a test is less than the level of significance, then which of these conclusions is correct?
A. The value of the test statistic is in the rejection region for this test.
B. The sample size should be increased to decrease the margin of error.
C. The null hypothesis is true.
D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis.
E. None of these is a valid conclusion.

59. C. The null hypothesis is true.
If the P-value of a test is less than the level of significance, then which of these conclusions is correct?
A. The value of the test statistic is in the rejection region for this test.
B. The sample size should be increased to decrease the margin of error.
C. The null hypothesis is true.
D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis.
E. None of these is a valid conclusion.

60. Problem # 1
A. If the P-value is less than α, then the result is “statistically significant.” Reject H0.
This corresponds to the test statistic falling in the rejection region.

61. If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider?
I. Increase the confidence level
II. Increase the sample size
III. Increase the margin of error
IV. Increase the probability of a Type I error

62. If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider?
I. Increase the confidence level
II. Increase the sample size
III. Increase the margin of error
IV. Increase the probability of a Type I error

63. One-sided or Two-sided?
Page 537
E64
E66
E67
E68

64. One-sided or Two-sided?
E64: Two-sided
E66:
E67:
E68:

65. One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67:
E68:

66. One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67: One-sided
E68:

67. One-sided or Two-sided?
E64: Two-sided
E66: Two-sided
E67: One-sided
E68: One-sided

68. Questions?