Questions: What would happen (other things being equal) to a confidence interval if you calculated a 99 percent confidence interval rather than a 95 percent confidence interval? The 99 percent confidence interval will a) be narrower b) not change c) be wider d) increase the value of your point estimate

Solution Steps

For a sample size \( n = 30 \) and a population standard deviation \( \sigma = 1.5 \), the 95% confidence interval is calculated using the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where \( z \) for a 95% confidence level is approximately \( 1.96 \). Thus, the calculation is:

\[ 0 \pm 1.96 \cdot \frac{1.5}{\sqrt{30}} = 0 \pm 1.96 \cdot 0.2739 \approx 0 \pm 0.5368 \]

This results in the 95% confidence interval:

\[ (-0.5368, 0.5368) \]

For the same sample size \( n = 30 \) and population standard deviation \( \sigma = 1.5 \), the 99% confidence interval is calculated using the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where \( z \) for a 99% confidence level is approximately \( 2.5758 \). Thus, the calculation is:

\[ 0 \pm 2.5758 \cdot \frac{1.5}{\sqrt{30}} = 0 \pm 2.5758 \cdot 0.2739 \approx 0 \pm 0.7054 \]

This results in the 99% confidence interval:

\[ (-0.7054, 0.7054) \]

The width of the 95% confidence interval is calculated as:

\[ \text{Width}_{95} = 0.5368 - (-0.5368) = 1.0736 \]

The width of the 99% confidence interval is calculated as:

\[ \text{Width}_{99} = 0.7054 - (-0.7054) = 1.4108 \]

Since \( \text{Width}_{99} = 1.4108 \) is greater than \( \text{Width}_{95} = 1.0736 \), we conclude that the 99% confidence interval is wider than the 95% confidence interval.

Final Answer