Concept explainers
One of the questions on a survey of 1000 adults asked if today’s children will be better off than their parents (Rasmussen Reports website, October 26, 2012). Representative data are shown in the DATAfile named ChildOutlook. A response of Yes indicates that the adult surveyed did think today’s children will be better off than their parents. A response of No indicates that the adult surveyed did not think today’s children will be better off than their parents. A response of Not Sure was given by 23% of the adults surveyed.
- a. What is the point estimate of the proportion of the population of adults who do think that today’s children will be better off than their parents?
- b. At 95% confidence, what is the margin of error?
- c. What is the 95% confidence interval for the proportion of adults who do think that today’s children will be better off than their parents?
- d. What is the 95% confidence interval for the proportion of adults who do not think that today’s children will be better off than their parents?
- e. Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?
a.
Find the point estimate of the proportion of the population of adults who think that today’s children will be better than their parents.
Answer to Problem 37E
The point estimate of the proportion of the population of adults who think that today’s children will be better than their parents is 0.24.
Explanation of Solution
Calculation:
The survey is on the today’s children will be better than their parents or not. The sample of 1,000 adults gave responses on the question. The responses of the survey is yes, no and not sure.
From the data, the response of yes is 240, the response of no is 530 and the not sure is 230.
The sampling distribution of
The value of
Thus, the point estimate of the proportion is 0.24.
b.
Find the margin of error at 95% confidence.
Answer to Problem 37E
The margin of error at 95% confidence is 0.0265.
Explanation of Solution
Calculation:
The formula for margin of error is,
From the results of part (a),
The value of
From the “Table 8.1 value of
The value of margin of error is,
Thus, the value of margin of error is 0.0265.
c.
Find the 95% confidence interval for the population proportion of the population of adults who think that today’s children will be better than their parents.
Answer to Problem 37E
The 95% confidence interval for the population proportion of the population of adults who think that today’s children will be better than their parents is (0.2135, 0.2665).
Explanation of Solution
Calculation:
From the results of parts (a) and (b), the point estimate of the proportion
The 95% confidence interval for population proportion is,
Thus, the 95% confidence interval for population proportion is (0.2135, 0.2665).
d.
Find the 95% confidence interval for the population proportion of the population of adults who do not think that today’s children will be better than their parents.
Answer to Problem 37E
The 95% confidence interval for the population proportion of the population of adults who do not think that today’s children will be better than their parents is (0.4991, 0.5609).
Explanation of Solution
Calculation:
The value of
Thus, the point estimate of the proportion is 0.53.
From the “Table 8.1 value of
The value of margin of error is,
Thus, the value of margin of error is 0.0309.
The 95% confidence interval for population proportion is,
Thus, the 95% confidence interval for population proportion is (0.4991, 0.5609).
e.
Find the smallest margin of error in part (c) and (d).
Explain the reason for smaller margin of error
Answer to Problem 37E
The smaller margin of error contains in part (c) confidence interval and the smaller margin of error because
Explanation of Solution
Calculation:
From the part (c), the margin of error is 0.0265.
From the part (d), the margin of error is 0.0309.
The smaller margin of error contains in part (c) confidence interval.
In part (c), the point estimator
Thus, the smaller margin of error because
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Chapter 8 Solutions
Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)
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