1. Inference for Proportions
Chapter 8
Inference for Proportions

2. Inference: using results from a random sample to draw conclusions about a population

3. In this chapter, you will use your understanding of sampling distributions developed in Chapter 7

4. One basic fact about sampling distributions will be used over and over again

5. If the sampling distribution can be considered approximately normal, 95% of all the sample means (or sample proportions) will fall within 1.96 standard errors of the population mean (or population proportion).

6. Estimating a Proportion with Confidence
Section 8.1
Estimating a Proportion with Confidence

7. Reasonably Likely and Rare Events
Reasonably likely events are those in the middle 95% of the distribution of all possible outcomes.
The outcomes in the upper 2.5% and lower 2.5% of the distribution are rare events - - they happen, but rarely.

8. Reasonably Likely and Rare Events
middle 95%

9. Reasonably Likely Events
Given:
(1) Random sampling from a binomial population is used repeatedly

10. Reasonably Likely Events
Given:
(1) Random sampling from a binomial population is used repeatedly
(2) Both np and n(1- p) are at least 10

11. Reasonably Likely Events
Given:
(1) Random sampling from a binomial population is used repeatedly
(2) Both np and n(1- p) are at least 10
-- have approx. normal distribution

12. Reasonably Likely Events
Given:
(1) Random sampling from a binomial population is used repeatedly
(2) Both np and n(1-p) are at least 10
Then, 95% of all sample proportions p will fall within 1.96 standard errors of the population proportion, p.

13. Reasonably Likely Events
or about 95% of sample proportions will fall within the interval
where n is the sample size.

14. Suppose you flip a fair coin 100 times and define heads as success .
(1) What are the reasonably likely values of the sample proportion p?
(2) What number of heads is reasonably likely?

15. Suppose you flip a fair coin 100 times.
(1) What are the reasonably likely values of the sample proportion p?
(2) What number of heads is reasonably likely?
What is p?
What is n?

16. Suppose you flip a fair coin 100 times.
(1) What are the reasonably likely values of the sample proportion p?
(2) What number of heads is reasonably likely?
What is p? probability of success is 0.5
What is n?

17. Suppose you flip a fair coin 100 times.
(1) What are the reasonably likely values of the sample proportion p?
(2) What number of heads is reasonably likely?
What is p? probability of success is 0.5
What is n? 100 flips means sample size is 100

18. Suppose you flip a fair coin 100 times.
(1) What are the reasonably likely values of the sample proportion p? =

19. Suppose you flip a fair coin 100 times.
(1) What are the reasonably likely values of the sample proportion p? =
So, reasonable likely values of p are from 0.4 to 0.6

20. Suppose you flip a fair coin 100 times.
(2) What number of heads is reasonably likely?

21. Suppose you flip a fair coin 100 times.
(2) What number of heads is reasonably likely?
In about 95% of the samples, the number of successes x in the sample will be in the interval
about 40 to 60 heads

22. Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance.
(a) There is a 95% chance that p will be between what two values?

23. Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance.

24. Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance.
(b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

25. (0.308, 0.392)
(b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

26. (b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

27. Suppose 40% of students in your graduating class plan to go on to higher education. You survey a random sample of 50 of your classmates and compute the sample proportion, p, of students who plan to go on to higher education.
(a) There is a 95% chance that p will be between what two numbers?
(b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education?

28. (a) There is a 95% chance that p will be between what two numbers?

29. (b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education?

30. (b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education?
(0.264, 0.536)
Getting 25 out of 50 is a sample
proportion of 0.5.
This is a reasonably likely event from a
population with p = 0.4.

31. Turn to page 470. Read through Activity 8.1a.

32. Activity 8.1a, Page 470
1. Out of a sample of 40 students, 27 students could make the Vulcan salute with both hands at once.
Write-up for this lab is due Monday.
Justify your answers—simple yes or no answers earn no credit

33. A 95% confidence interval consists of those population proportions p for which the sample proportion p is reasonably likely.

34. A Complete Chart of Reasonably Likely Sample Proportions for n = 40

35. A Complete Chart of Reasonably Likely Sample Proportions for n = 40

36. Page 473, D3

37. Page 473, D3

38. Page 473, D3

39. Page 473, D3

40. Page 473, D4

41. Page 473, D4

43. Page 473, D4 No. The horizontal line segment at p = 0.3
goes from about to 0.442, so a
sample proportion of 0.6 isn’t a reasonably
likely result for a population with only 30%
men.

44. Page 473, D5

45. Page 473, D5

46. Page 473, D5

47. Page 473, D5

48. Page 473, D6

49. Page 473, D6 The populations for which a sample
proportion of 0.5 is reasonably likely are
35% to 65%. This can be written as 50%
15%.

50. Page 473, D7

51. Page 473, D7 You don’t need a confidence interval for p
because you already know exactly what that is
from our sample and you know that it probably
would have been different if you had taken a
different sample.
What you want is an interval that has a good
chance of capturing the true but unknown
proportion of successes p in the population from
which the sample was taken.

52. A confidence interval for the proportion of successes p in the population is given by the formula
Here n is the sample size and p is the proportion of successes in the sample.

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

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

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

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

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

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

59. 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

60. 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

61. The quantity
is called the margin of error.

62. The quantity
is called the margin of error.

63. Questions?