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Chapter 8: Problem 14

A simple random sample with \(n=54\) provided a sample mean of 22.5 and a sample standard deviation of 4.4 a. Develop a \(90 \%\) confidence interval for the population mean. b. Develop a \(95 \%\) confidence interval for the population mean. c. Develop a \(99 \%\) confidence interval for the population mean. d. What happens to the margin of error and the confidence interval as the confidence level is increased?

Short Answer

Expert verified
a. The 90% confidence interval for the population mean is approximately (21.51, 23.49). b. The 95% confidence interval for the population mean is approximately (21.39, 23.61). c. The 99% confidence interval for the population mean is approximately (21.19, 23.81). d. As the confidence level increases, the margin of error and the width of the confidence interval also increase, providing a less precise but more confident estimate of the population mean.

Step by step solution

01

Calculate the margin of error

To calculate the margin of error, we need to find the corresponding z-scores for the given confidence levels (90%, 95%, and 99%). We can find the z-scores in a standard normal distribution z-table or by using statistical software. For a 90% confidence level, the z-score is 1.645. For a 95% confidence level, the z-score is 1.96. For a 99% confidence level, the z-score is 2.576. Now, we calculate the margin of error: \(E_{90} = 1.645 \cdot \frac{4.4}{\sqrt{54}}\) \(E_{95} = 1.96 \cdot \frac{4.4}{\sqrt{54}}\) \(E_{99} = 2.576 \cdot \frac{4.4}{\sqrt{54}}\)
02

Calculate the confidence intervals

We will now use the calculated margin of errors and the sample mean (22.5) to find the confidence intervals for each given confidence level: \(Confidence \, Interval_{90} = 22.5 \pm E_{90}\) \(Confidence \, Interval_{95} = 22.5 \pm E_{95}\) \(Confidence \, Interval_{99} = 22.5 \pm E_{99}\)
03

Interpret the results

After calculating the confidence intervals, we can conclude the following: a. At a 90% confidence level, the confidence interval for the population mean is approximately (21.51, 23.49). b. At a 95% confidence level, the confidence interval for the population mean is approximately (21.39, 23.61). c. At a 99% confidence level, the confidence interval for the population mean is approximately (21.19, 23.81).
04

Analyze the effects of increasing the confidence level on the margin of error and confidence interval

As we can observe from the results in Steps 1 and 2: - As the confidence level increases, the margin of error (E) increases (E_90 < E_95 < E_99). - As the margin of error increases, the width of the confidence interval also increases. In conclusion, as the confidence level is increased, the margin of error becomes larger, and the confidence interval becomes wider. This means that we are more confident in our estimate of the population mean, but at the cost of a less precise estimate.

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Most popular questions from this chapter

In a survey, the planning value for the population proportion is \(p^{*}=.35 .\) How large a sample should be taken to provide a \(95 \%\) confidence interval with a margin of error of \(.05 ?\)

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