Concept explainers
(a)
To test: Whether an 99% confidence interval would be wider or narrower than the 95% confidence interval obtained in Exercise 8.31.
(a)
Answer to Problem 32E
Solution: The 99% confidence interval would be wider. An 99% confidence interval is obtained as
Explanation of Solution
Calculation: The 95% confidence interval is obtained as
The formula for 99% confidence interval for population proportion p is defined as:
Where,
The sample proportion is provided as:
Therefore, the sample proportion
The formula for margin of error m is defined as:
In the above formula,
The formula for standard error
The sample proportion
Therefore, the standard error is obtained as 0.0131. The value of
So, the margin of error is obtained as:
Therefore, the margin of error is obtained as 0.033798.
Substitute the obtained values of margin of error and sample proportion in the formula for confidence interval. Therefore, an 99% confidence interval is obtained as:
Therefore, an 99% confidence interval is obtained as
The width of the 99% confidence interval is obtained as:
The width of the 95% confidence interval is obtained as:
Conclusion: The obtained widths of the two confidence levels show that the 99% confidence interval is wider than the 95% confidence interval.
(b)
To test: Whether a 90% confidence interval would be wider or narrower than the 95% confidence interval obtained in Exercise 8.31.
(b)
Answer to Problem 32E
Solution: A 90% confidence interval would be wider. A 90% confidence interval is obtained as
Explanation of Solution
Calculation: The 95% confidence interval is obtained as
The formula for 90% confidence interval for p is defined as:
The sample proportion is provided as:
Therefore, the sample proportion
The formula for margin of error m is defined as:
In the above formula,
The formula for standard error
The sample proportion
Therefore, the standard error is obtained as 0.0131. The value of
So, the margin of error is obtained as:
Therefore, the margin of error is obtained as 0.02155.
Substitute the obtained values of margin of error and sample proportion in the formula for confidence interval. Therefore, a 90% confidence interval is obtained as:
Therefore, a 90% confidence interval is obtained as
The width of the 90% confidence interval is obtained as:
The width of the 95% confidence interval is obtained as:
Conclusion: The obtained widths of the two confidence levels show that 90% confidence interval is narrower than the 95% confidence interval.
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Chapter 8 Solutions
INTRO.TO PRACTICE STATISTICS-ACCESS
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