Use the standard normal distribution or the $t$-distribution to construct a $99 \%$ confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results. In a random sample of 19 mortgage institutions, the mean interest rate was $3.41 \%$ and the standard deviation was $0.36 \%$. Assume the interest rates are normally distributed. B. Neither distribution can be used to construct the confidence interval. Interpret the results. Choose the correct answer below. A. If a large sample of institutions are taken approximately $99 \%$ of them will have an interest rate between the bounds of the confidence interval. B. With $99 \%$ confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval. C. It can be said that $99 \%$ of institutions have an interest rate between the bounds of the confidence interval. D. Neither distribution can be used to construct the confidence interval.

Step 1 ：The problem is asking to construct a 99% confidence interval for the population mean. We are given a sample size of 19, a sample mean of 3.41%, and a sample standard deviation of 0.36%. We are also told to assume the interest rates are normally distributed.

Step 2 ：Since the sample size is less than 30, we should use the t-distribution to construct the confidence interval. The t-distribution is more appropriate for smaller sample sizes, while the standard normal distribution is more appropriate for larger sample sizes.

Step 3 ：The formula for a confidence interval is: \[\bar{x} \pm t \left( \frac{s}{\sqrt{n}} \right)\] where \(\bar{x}\) is the sample mean, \(t\) is the t-score corresponding to the desired level of confidence, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 4 ：Using the given values, we find that the t-score is approximately 2.878 and the margin of error is approximately 0.238.

Step 5 ：Substituting these values into the formula, we find that the lower bound of the confidence interval is approximately 3.17% and the upper bound is approximately 3.65%.

Step 6 ：This means that we are 99% confident that the true population mean interest rate lies between 3.17% and 3.65%.

Step 7 ：Therefore, the correct interpretation of the results is: With 99% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.

Step 8 ：Final Answer: \(\boxed{\text{B. With 99% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.}}\)