Question

question 3 of 10
a 95% confidence interval is found to be (24, 30). the sample standard deviation is 7.5, and the sample size is 25. if you wanted a smaller interval, what could you do?

a. increase the sample size to 36.
b. decrease the sample size to 20.
c. none of these
d. increase the confidence level to 99.7%.

Explanation:

Step1: Recall Confidence Interval Formula

For a confidence interval (assuming z - interval or t - interval, the margin of error \(E\) is related to sample size \(n\), standard deviation \(\sigma\) (or \(s\) for sample), and confidence level. The formula for margin of error (for z - interval, approximate) is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) (for t - interval, \(E=t_{\alpha/2,df}\frac{s}{\sqrt{n}}\)). The width of the confidence interval is \(2E\). To decrease the width (make the interval smaller), we need to decrease \(E\).

Step2: Analyze Effect of Sample Size

From the formula \(E\propto\frac{1}{\sqrt{n}}\) (inversely proportional to the square root of sample size). If we increase \(n\), \(\sqrt{n}\) increases, so \(E\) decreases. If we decrease \(n\), \(\sqrt{n}\) decreases, so \(E\) increases (interval becomes wider). So increasing sample size (e.g., from 25 to 36) will decrease the margin of error and thus the width of the confidence interval.

Step3: Analyze Effect of Confidence Level

The critical value \(z_{\alpha/2}\) (or \(t_{\alpha/2}\)) increases as the confidence level increases (e.g., 95% to 99.7%). Since \(E\propto z_{\alpha/2}\) (or \(t_{\alpha/2}\)), increasing the confidence level will increase \(E\) (interval becomes wider). Decreasing the confidence level would decrease \(E\), but option D is increasing the confidence level, so D is incorrect.

Step4: Evaluate Options

• Option A: Increase sample size to 36. Since \(n\) increases, \(E\) decreases, so the interval becomes smaller. This is correct.
• Option B: Decrease sample size to 20. This will increase \(E\), making the interval wider. Incorrect.
• Option D: Increase confidence level to 99.7%. This increases \(E\), making the interval wider. Incorrect.
• Option C: Not applicable as A is correct.

Answer:

A. Increase the sample size to 36.