1. STATISTICS INFORMED DECISIONS USING DATA
Fifth Edition
Chapter 9
Estimating the Value of a Parameter
Copyright © 2018, 2014, 2011 Pearson Education, Inc. All Rights Reserved

2. 9.1 Estimating a Population Proportion Learning Objectives
1. Obtain a point estimate for the population proportion 2. Construct and interpret a confidence interval for the population proportion 3. Determine the sample size necessary for estimating a population proportion within a specified margin of error

3. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Obtain a Point Estimate for the Population Proportion (1 of 3)
A point estimate is the value of a statistic that estimates the value of a parameter.

4. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Obtain a Point Estimate for the Population Proportion (2 of 3)
Parallel Example 1: Calculating a Point Estimate for the Population Proportion
In July of 2008, a Quinnipiac University Poll asked 1783 registered voters nationwide whether they favored or opposed the death penalty for persons convicted of murder were in favor.
Obtain a point estimate for the proportion of registered voters nationwide who are in favor of the death penalty for persons convicted of murder.

5. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Obtain a Point Estimate for the Population Proportion (3 of 3)
Solution
Obtain a point estimate for the proportion of registered voters nationwide who are in favor of the death penalty for persons convicted of murder.

6. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (1 of 11)
A confidence interval for an unknown parameter consists of an interval of numbers based on a point estimate. The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted (1 − α) · 100%.

7. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (2 of 11)
For example, a 95% level of confidence (α = 0.05) implies that if 100 different confidence intervals are constructed, each based on a different sample from the same population, we will expect 95 of the intervals to contain the parameter and 5 not to include the parameter.

8. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (3 of 11)
Confidence interval estimates for the population proportion are of the form Point estimate ± margin of error. The margin of error of a confidence interval estimate of a parameter is a measure of how accurate the point estimate is.

9. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (4 of 11)
The margin of error depends on three factors:
Level of confidence: As the level of confidence increases, the margin of error also increases.
Sample size: As the size of the random sample increases, the margin of error decreases.
Standard deviation of the population: The more spread there is in the population, the wider our interval will be for a given level of confidence.

10. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (5 of 11)
NOTE: We also require that each trial be independent; when sampling from finite populations (n ≤ 0.05N).

11. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (6 of 11)
Interpretation of a Confidence Interval
A (1 − α) · 100% confidence interval indicates that (1 − α) · 100% of all simple random samples of size n from the population whose parameter is unknown will result in an interval that contains the parameter.

12. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (7 of 11)
Constructing a (1 − α) · 100% Confidence Interval for a Population Proportion Suppose that a simple random sample of size n is taken from a population. A (1 − α) · 100% confidence interval for p is given by the following quantities

13. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (8 of 11)
Margin of Error
The margin of error, E, in a (1 − α) · 100% confidence interval for a population proportion is given by

14. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (9 of 11)
Parallel Example 2: Constructing a Confidence Interval for a Population Proportion
In July of 2008, a Quinnipiac University Poll asked 1783 registered voters nationwide whether they favored or opposed the death penalty for persons convicted of murder were in favor.
Obtain a 90% confidence interval for the proportion of registered voters nationwide who are in favor of the death penalty for persons convicted of murder.

15. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (10 of 11)
Solution

16. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Construct and Interpret a Confidence Interval for the Population Proportion (11 of 11)
Solution
We are 90% confident that the proportion of registered voters who are in favor of the death penalty for those convicted of murder is between 0.61 and 0.65.

17. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (1 of 6)
Sample size needed for a specified margin of error, E, and level of confidence (1 − α):

18. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (2 of 6)
Two possible solutions:

19. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (3 of 6)
Sample Size Needed for Estimating the Population Proportion p
The sample size required to obtain a (1 − α) · 100% confidence interval for p with a margin of error E is given by

20. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (4 of 6)
Sample Size Needed for Estimating the Population Proportion p
If a prior estimate of p is unavailable, the sample size required is

21. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (5 of 6)
Parallel Example 4: Determining Sample Size
A sociologist wanted to determine the percentage of residents of America that only speak English at home. What size sample should be obtained if she wishes her estimate to be within 3 percentage points with 90% confidence assuming she uses the estimate obtained from the Census 2000 Supplementary Survey of 82.4%?

22. 9. 1 Estimating a Population Proportion 9. 1
9.1 Estimating a Population Proportion Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error (6 of 6)
We round this value up to 437. The sociologist must survey 437 randomly selected American residents.

23. 9.2 Estimating a Population Mean Learning Objectives
1. Obtain a point estimate for the population mean 2. State properties of Student’s t-distribution 3. Determine t-values 4. Construct and interpret a confidence interval for a population mean 5. Determine the sample size needed to estimate the population mean within a specified margin of error

24. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Obtain a Point Estimate for the Population Mean (1 of 3)
A point estimate is the value of a statistic that estimates the value of a parameter.

25. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Obtain a Point Estimate for the Population Mean (2 of 3)
Parallel Example 1: Computing a Point Estimate
Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent the weights (in grams) of 17 randomly selected pennies minted after 1982.
Treat the data as a simple random sample. Estimate the population mean weight of pennies minted after 1982.

26. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Obtain a Point Estimate for the Population Mean (3 of 3)
Solution
The sample mean is

27. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (1 of 9)
Student’s t-Distribution
Suppose that a simple random sample of size n is taken from a population. If the population from which the sample is drawn follows a normal distribution, the distribution of

28. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (2 of 9)
Parallel Example 2: Comparing the Standard Normal Distribution to the t-Distribution Using Simulation
Obtain 1,000 simple random samples of size n = 5 from a normal population with μ = 50 and σ = 10.
Determine the sample mean and sample standard deviation for each of the samples.

29. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (3 of 9)
Histogram for z

30. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (4 of 9)
Histogram for t

31. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (5 of 9)
CONCLUSION:
The histogram for z is symmetric and bell-shaped with the center of the distribution at 0 and virtually all the rectangles between −3 and 3. In other words, z follows a standard normal distribution.

32. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (6 of 9)
CONCLUSION:
The histogram for t is also symmetric and bell-shaped with the center of the distribution at 0, but the distribution of t has longer tails (i.e., t is more dispersed), so it is unlikely that t follows a standard normal distribution. The additional spread in the distribution of t can be attributed to the fact that we use s to find t instead of σ. Because the sample standard deviation is itself a random variable (rather than a constant such as σ), we have more dispersion in the distribution of t.

33. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (7 of 9)
Properties of the t-Distribution
1. The t-distribution is different for different degrees of freedom.
2. The t-distribution is centered at 0 and is symmetric about 0.

34. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (8 of 9)
Properties of the t-Distribution
5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further variability into the t- statistic.
6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because, as the sample size n increases, the values of s get closer to the values of σ, by the Law of Large Numbers.

35. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean State Properties of Student’s t-Distribution (9 of 9)

36. 9.2 Estimating a Population Mean 9.2.3 Determine t-Values (1 of 3)
The notation tα is the t-value such that the area under the standard normal curve to the right is α. The figure illustrates the notation.

37. 9.2 Estimating a Population Mean 9.2.3 Determine t-Values (2 of 3)
Parallel Example 3: Finding t-values
Find the t-value such that the area under the t-distribution to the right of the t-value is 0.2 assuming 10 degrees of freedom. That is, find t0.20 with 10 degrees of freedom.

38. 9.2 Estimating a Population Mean 9.2.3 Determine t-Values (3 of 3)
Solution
The figure to the right shows the graph of the t-distribution with 10 degrees of freedom.

39. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (1 of 11)
Constructing a (1 − α)100% Confidence Interval for μ
Provided
sample data come from a simple random sample or randomized experiment,
sample size is small relative to the population size (n ≤ 0.05N), and
the data come from a population that is normally distributed, or the sample size is large.

40. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (2 of 11)
Constructing a (1 − α)100% Confidence Interval for μ
A (1 − α) · 100% confidence interval for μ is given by

41. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (3 of 11)
Parallel Example: Using Simulation to Demonstrate the Idea of a Confidence Interval
We will use Minitab to simulate obtaining 30 simple random samples of size n = 8 from a population that is normally distributed with μ = 50 and σ = 10. Construct a 95% confidence interval for each sample. How many of the samples result in intervals that contain μ = 50?

42. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (4 of 11)
SAMPLE
MEAN
95.0% CI C1
47.07
(40.14, 54.00) C2
49.33
(42.40, 56.26) C3
50.62
(43.69, 57.54) C4
47.91
(40.98, 54.84) C5
44.31
(37.38, 51.24) C6
51.50
(44.57, 58.43) C7
52.47
(45.54, 59.40) C8
59.62
(52.69, 66.54) C9
43.49
(36.56, 50.42)
C10
55.45
(48.52, 62.38)

43. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (5 of 11)
SAMPLE
MEAN
95.0% CI
C11
50.08
(43.15, 57.01)
C12
56.37
(49.44, 63.30)
C13
49.05
(42.12, 55.98)
C14
47.34
(40.41, 54.27)
C15
50.33
(43.40, 57.25)
C16
44.81
(37.88, 51.74)
C17
51.05
(44.12, 57.98)
C18
43.91
(36.98, 50.84)
C19
46.50
(39.57, 53.43)
C20
49.79
(42.86, 56.72)

44. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (6 of 11)
SAMPLE
MEAN
95.0% CI
C21
48.75
(41.82, 55.68)
C22
51.27
(44.34, 58.20)
C23
47.80
(40.87, 54.73)
C24
56.60
(49.67, 63.52)
C25
47.70
(40.77, 54.63)
C26
51.58
(44.65, 58.51)
C27
47.37
(40.44, 54.30)
C28
61.42
(54.49, 68.35)
C29
46.89
(39.96, 53.82)
C30
51.92
(44.99, 58.85)

45. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (7 of 11)
Note that 28 out of 30, or 93%, of the confidence intervals contain the population mean μ = 50. In general, for a 95% confidence interval, any sample mean that lies within 1.96 standard errors of the population mean will result in a confidence interval that contains μ.

46. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (8 of 11)
Parallel Example 4: Constructing a Confidence Interval about a Population Mean
Construct a 99% confidence interval about the population mean weight (in grams) of pennies minted after Assume μ = 0.02 grams.

47. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (9 of 11)

48. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (10 of 11)

49. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Construct and Interpret a Confidence Interval for the Population Mean (11 of 11)
We are 99% confident that the mean weight of pennies minted after 1982 is between and grams.

50. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Determine the Sample Size Needed to Estimate the Population Mean within a Specified Margin of Error (1 of 3)
Determining the Sample Size n
The sample size required to estimate the population mean, µ, with a level of confidence (1 − α)·100% with a specified margin of error, E, is given by

51. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Determine the Sample Size Needed to Estimate the Population Mean within a Specified Margin of Error (2 of 3)
Parallel Example 7: Determining the Sample Size
Back to the pennies. How large a sample would be required to estimate the mean weight of a penny manufactured after within grams with 99% confidence? Assume s = 0.02.

52. 9. 2 Estimating a Population Mean 9. 2
9.2 Estimating a Population Mean Determine the Sample Size Needed to Estimate the Population Mean within a Specified Margin of Error (3 of 3)
Rounding up, we find n = 107.

53. 9.3 Putting It Together: Which Procedure Do I Use? Learning Objective
1. Determine the appropriate confidence interval to construct

54. 9. 3 Putting It Together: Which Procedure Do I Use. 9. 3
9.3 Putting It Together: Which Procedure Do I Use? Determine the Appropriate Confidence Interval to Construct