1. Sampling Distribution of the Sample Proportion
Section 7.3
Sampling Distribution of the Sample Proportion

2. Binomial Probability Distribution
Recall, in Chapter 6 we discussed a binomial probability distribution.
What conditions must be satisfied to have a binomial probability distribution?

3. Binomial Probability Distribution
A binomial probability distribution must satisfy these conditions: B I N S

4. Binomial Probability Distribution
B: binomial –each trial must have one of two outcomes—”success” or “failure” I N S

5. Binomial Probability Distribution
B: binomial –each trial must have one of two outcomes—”success” or “failure”
I: each trial is independent of the others N: S

6. Binomial Probability Distribution
B: binomial –each trial must have one of two outcomes—”success” or “failure”
I: each trial is independent of the others
N: there is a fixed number, n, of trials S

7. Binomial Probability Distribution
B: binomial –each trial must have one of two outcomes—”success” or “failure”
I: each trial is independent of the others
N: there is a fixed number, n, of trials
S: P(success) does not change

8. Properties of the Sampling Distribution of the Number of Successes
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:

9. Properties of the Sampling Distribution of the Number of Successes
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:
has mean x = np

10. Properties of the Sampling Distribution of the Number of Successes
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:
has mean x = np
has standard error

11. Properties of the Sampling Distribution of the Number of Successes
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:
has mean x = np
has standard error
will be approximately normal as long as n is large enough

12. How Large is Large Enough?
As a conservative guideline, if both

13. How Large is Large Enough?
As a conservative guideline, if both np and n(1 - p) are at least 10,

14. How Large is Large Enough?
As a conservative guideline, if both np and n(1 - p) are at least 10, then using the normal distribution as an approximation for the shape of the sampling distribution will give reasonably accurate results.

15. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.

16. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.
(a) Suppose you take a random sample of 40 Mississipians.
How many do you expect will wear seat belts?

17. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.
Suppose you take a random sample of 40 Mississipians. How many do you expect will wear seat belts?
You expect that 60% of the 40, or 24, Mississipians will be wearing seat belts.

18. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.
(b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?

19. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.
(b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?
Hint: Can you use the normal approximation to the binomial distribution?

20. The use of seat belts continues to rise in the U. S
The use of seat belts continues to rise in the U.S., with overall seat belt usage of 82%. Mississippi lags behind the rest of the nation – only about 60% wear seat belts.
(b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?
np = 40(0.6) = 24
n(1 – p) = 40 (1 – 0.6) = 16
Since both are at least 10, you can use the normal approximation to the binomial distribution to determine the probability.

21. (b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?
P(30 or more) = normalcdf (lower bound, upper bound, x, x)

22. (b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?
P(30 or more) = normalcdf (lower bound, upper bound, x, x)
P(30 or more) = normalcdf (lower bound, upper bound, np, )

23. (b) What is the probability that 30 or more of the people in the sample of 40 wear seat belts?
P(30 or more) = normalcdf (lower bound, upper bound, x, x)
P(30 or more) = normalcdf (lower bound, upper bound, np, )
P(30 or more) = normalcdf(30, 1E99, 24, 3.098) ≈

24. Sampling Distribution of the Number of Successes
A survey of hundreds of thousands of college freshmen found that 63% believe “dissent is a critical component of the political process.”

25. Sampling Distribution of the Number of Successes
A survey of hundreds of thousands of college freshmen found that 63% believe “dissent is a critical component of the political process.”
Suppose you take a random sample of 100 of the freshmen surveyed. What is the probability that you will find that between 56 and 70 of the freshmen in your sample believe this?

26. Use Normal Distribution?
np = 100(0.63) = 63
n(1 – p) = 100 ( 1 – 0.63) = 37
Since both are at least 10, the shape of the sampling distribution will be approximately normal.
Now find x and x

27. P(between 56 and 70) = normalcdf(lower bound, upper bound, x, x)

28. P(between 56 and 70) = normalcdf(56, 70, 63, 4.83) ≈ 0.853
Therefore, there is about an 85.3% probability that a sample of 100 freshmen will contain between 56 and 70 freshmen who believe that dissent is a critical component of the political process.

29. Properties of the Sampling Distribution of the Sample Proportion
To change from number of successes to proportion of successes, divide by sample size, n.

30. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:

31. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
Mean of the sampling distribution is equal to the mean of the population, or

32. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p:
Standard error of the sampling distribution is equal to the standard deviation of the population divided by the square root of the sample size:

33. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:

34. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
As the sample size gets larger, the shape of the sampling distribution becomes more normal and

35. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
As the sample size gets larger, the shape of the sampling distribution becomes more normal and will be approximately normal if n is large enough (both np and n(1 – p) are at least 10).

36. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.

37. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Describe the shape, center, and spread of the sampling distribution of the proportion of drivers who would fail the test in a random sample of 60 drivers from these states.

38. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Random sample of 60 drivers from these states:
np = 60(.2) = 12
n(1 – p) = 60 (1 - .2) = 48
Both at least 10 so,
shape of sampling distribution of sample proportion is approximately normal.

39. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Random sample of 60 drivers from these states:
Mean of the sampling distribution is equal to the mean of the population = 0.2

40. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Random sample of 60 drivers from these states:
Spread: =
≈ 0.05

41. What happens if we quadruple the sample size?
Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Random sample of 60 drivers from these states:
Spread: =
≈ 0.05
What happens if we quadruple the sample size?

42. Drivers in the Northeast and Mid-Atlantic states had the highest failure rate, 20%, on the GMAC Insurance National Driver’s Test.
Random sample of 60 drivers from these states:
Spread: =
≈ 0.05
What happens if we quadruple the sample size? Spread is reduced in half.

43. What are the reasonably likely proportions of drivers in the sample who would fail the test?

44. mean ± 1.96(standard error)
What are the reasonably likely proportions of drivers who would fail the test?
mean ± 1.96(standard error)

45. mean ± 1.96(standard error) 0.2 ± 1.96(0.05)
What are the reasonably likely proportions of drivers who would fail the test?
mean ± 1.96(standard error)
0.2 ± 1.96(0.05)
So, reasonably likely proportions would be between about 0.1 and 0.3

46. In the 2000 U. S. Census, 53% of the population over age 30 were women
In the 2000 U.S. Census, 53% of the population over age 30 were women. Describe the shape, mean, and standard error of the sampling distribution of the sample proportion for random samples of size 100 taken from this population. Make an accurate sketch, with a scale on the horizontal axis of this distribution.

49. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
Use mean and standard error from previous problem.

50. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(lower bound, upper bound, mean, standard error)

51. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99, 0.09, 0.53, 0.05) = ?

52. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99 , 0.09, 0.53, 0.05) = 0
Reasonably likely event or not?

53. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99 , 0.09, 0.53, 0.05) = 0
The probability of getting 9 or fewer women just by chance is 0 so this is not a reasonably likely event.

54. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
of 40 drivers?
of 100 drivers?
of 400 drivers?

55. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
of 40 drivers?
of 100 drivers?
of 400 drivers?
Because the sampling distributions are approximately normal, in each case 95% of the potential values of the sample proportion will lie within 1.96 standard errors of the mean

56. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
a. of 40 drivers? b. of 100 drivers? c. of 400 drivers?

57. Questions?