1. Section 8.1
Day 2

2. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

3. According to a recent report, about 30% of 20- and 21-year-olds are enrolled in school. How many enrolled students is it reasonably likely to get in a sample of 40 randomly chosen 20- and 21-year-olds?

4. According to a recent report, about 30% of 20- and 21-year-olds are enrolled in school. How many enrolled students is it reasonably likely to get in a sample of 40 randomly chosen 20- and 21-year-olds?
Population sample

5. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

6. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

7. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)
6 to 18 students

8. Suppose that in a random sample of 40 adults, 26 of the adults have never married. Find the 95% confidence interval for the proportion of all adults who have never married.

9. Suppose that in a random sample of 40 adults, 26 of the adults have never married. Find the 95% confidence interval for the proportion of all adults who have never married.
Sample population

10. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

11. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

12. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)
(0.5, 0.8)

13. A confidence interval (CI) for the proportion of successes p in the population is given by the formula
Here n is the sample size, p is the proportion of successes in the sample, and z* is the critical value.

14. Value of z* depends on how confident you want to be that p will be in the confidence interval.

15. Value of z* depends on how confident you want to be that p will be in the confidence interval.
For 90% confidence interval, use z* =1.645

16. Value of z* depends on how confident you want to be that p will be in the confidence interval.
For 90% confidence interval, use z* = 1.645
For 95% confidence interval, use z* = 1.96

17. Value of z* depends on how confident you want to be that p will be in the confidence interval.
For 90% confidence interval, use z* = 1.645
For 95% confidence interval, use z* = 1.96
For 99% confidence interval, use z* = 2.576

18. As we become more confident, what happens to the width of the CI?
For 90% confidence interval, use z* = 1.645
For 95% confidence interval, use z* = 1.96
For 99% confidence interval, use z* = 2.576

19. Check Conditions
This confidence interval is reasonably accurate when three conditions are met:

20. Check Conditions
This confidence interval is reasonably accurate when three conditions are met:
(1) Sample was a simple random sample from a binomial population

21. Check Conditions
This confidence interval is reasonably accurate when three conditions are met:
(1) Sample was a simple random sample from a binomial population
(2) Both np and n(1 – p) are at least 10

22. Check Conditions
This confidence interval is reasonably accurate when three conditions are met:
(1) Sample was a simple random sample from a binomial population
(2) Both np and n(1 – p) are at least 10
(3) Size of the population is at least 10 times the size of the sample

23. A confidence interval for the proportion of successes p in the population is given by the formula

24. A confidence interval for the proportion of successes p in the population is given by the formula

25. A confidence interval for the proportion of successes p in the population is given by the formula
Standard error

26. The quantity
is called the margin of error.

27. The quantity
is called the margin of error.

28. Why is it true that when everything else is held constant, the margin of error is smaller with a larger sample size than with a smaller sample size?

29. Suppose you take a random sample and get p = 0.7.
a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error.

30. Suppose you take a random sample and get p = 0.7.
a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error.

31. Suppose you take a random sample and get p = 0.7.
a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error.
≈ 0.7 ±
So, the 95% CI is (0.6101, ) and the margin of error is approximately

32. Suppose you take a random sample and get p = 0.7.
b) What happens to the confidence interval and margin of error if you quadruple the sample size to 400?

33. Suppose you take a random sample and get p = 0.7.
b) What happens to the confidence interval and margin of error if you quadruple the sample size to 400?
The 95% CI is (0.6551, ) and the margin of error is approx

34. For n = 100: The 95% CI is (0.6101, 0.7898) and the margin of error is approximately 0.0898
Thus, quadrupling the sample size cuts the margin of error in half.

35. How does a confidence interval for large sample sizes compare to those for small sample sizes?

36. How does a confidence interval for large sample sizes compare to those for small sample sizes?
CI’s for large sample sizes are narrower than those for small sample sizes.
Why should this make sense to us?

37. CI’s for large sample sizes are narrower than those for small sample sizes.
This makes sense as the larger the sample size, the closer p should be to p.

38. What sample size should you use?

39. What sample size should you use?
The larger the sample size, the more precise the results will be.

40. What sample size should you use?
The larger the sample size, the more precise the results will be.
If larger is “better”, why limit our sample size?

41. Researchers always have limited time and money.
For practical reasons, we have to limit the size of our samples.

42. Researchers always have limited time and money.
For practical reasons, we have to limit the size of our samples.
So, how large is sufficient?

43. How large is sufficient depends on what margin of error is acceptable to you.
Solve this formula for n.
n = ?

46. To use this formula, you need three pieces of information.

47. To use this formula, you need three pieces of information:
(1) you have to know the margin of error that is acceptable

48. To use this formula, you need three pieces of information:
(1) you have to know the margin of error that is acceptable
(2) you have to decide on a level of confidence so you know what value of z* to use

49. To use this formula, you need three pieces of information:
(1) you have to know the margin of error that is acceptable
(2) you have to decide on a level of confidence so you know what value of z* to use
(3) you have to have an estimate of p.

50. To use this formula, you need three pieces of information:
(1) you have to know the margin of error that is acceptable
(2) you have to decide on a level of confidence so you know what value of z* to use
(3) you have to have an estimate of p.
--if you have a good estimate of p, use it in this formula

51. To use this formula, you need three pieces of information:
(1) you have to know the margin of error that is acceptable
(2) you have to decide on a level of confidence so you know what value of z* to use
(3) you have to have an estimate of p.
--if you have a good estimate of p, use it in this formula
--if you do not have a good estimate of p, use p = 0.5

52. What sample size should you use for a survey if you want the margin of error to be at most 3% with 95% confidence but you have no estimate of p?

53. What sample size should you use for a survey if you want the margin of error to be at most 3% with 95% confidence but you have no estimate of p?
Because no estimate of p, use p = 0.5

54. n = (1.96)2

55. You should use a sample size of 1068.
Impossible to use and 1067 would be too small.

56. Page 481, D17
Suppose it costs $5 to survey each person in your sample. You estimate that p is about 0.5.
What will your survey cost if you want a 95% confidence interval with a margin of error of:
About 10%? About 1%? About 0.1%?

60. Page 484, P11(a)
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).
(a) Check to see if the three conditions for computing a confidence interval are met in this case. (Need to be specific, not just general here)

61. Page 484, P11(a) The problem states that the sample was
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).
The problem states that the sample was
selected randomly from all U.S. teens aged
13 to 17.

62. Page 484, P11(a)
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).

63. Page 484, P11 There are well over 600(10) = 6000 teens
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).
There are well over 600(10) = 6000 teens
aged 13 to 17 in the United States.
All three conditions are met.

64. Page 484, P11(b)
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).
(b) Find a 95% confidence interval for the percentage of all teenagers in the United States who would give their school a D rating.

65. Page 484, P11(b)
In a survey consisting of a randomly selected national sample of 600 teens ages 13-17, 4% of the 600 students responding gave their school a D rating (on a scale of A, B, C, D, F).
(b) Find a 95% confidence interval for the percentage of all teenagers in the United States who would give their school a D rating.

66. Page 487, E8
Conditions met?

67. Page 487, E8
(1) There is no indication of how the sample was selected, so the random sample condition may not have been met

68. Page 487, E8
(1) There is no indication of how the sample was selected, so the random sample condition may not have been met
(2) np = 885(0.40) = 354 and n(1 – p) = 885(0.6) = 531 are both at least 10

69. Page 487, E8
(1) There is no indication of how the sample was selected, so the random sample condition may not have been met
(2) np = 885(0.40) = 354 and n(1 – p) = 885(0.6) = 531 are both at least 10
(3) There are more than 885(10) = 8850 adults nationwide
So, two of the three conditions are met

70. Page 487, E8
90% confidence interval is?

71. Page 487, E8 90% confidence interval
you can have confidence in this interval only
if the sampling was done randomly.

72. Interpreting Confidence Interval
Two parts:

73. Interpreting Confidence Interval
Two parts:
(1) Describing what is in the confidence interval

74. Interpreting Confidence Interval
Two parts:
(1) Describing what is in the confidence interval
(2) Describing what is meant by “confidence”

75. 95% Confidence Interval
(1) Describing what is in the confidence interval
You are 95% confident that if you could examine each unit in the population, the true proportion of successes, p, in this population would be in this confidence interval.

76. 95% Confidence Interval (2) Describing what is meant by “confidence”
If you were to repeat this process 100 times and construct the 100 resulting confidence intervals, you would expect 95 of them to contain the population proportion of success, p.
In other words, 95% of the time the process results in a confidence interval that captures the true value of p.

77. Questions?