Step 1 :The problem provides us with a sample of 16 mortgage institutions with a mean interest rate of 3.69% and a standard deviation of 0.33%. The interest rates are assumed to be normally distributed. We are asked to determine which distribution should be used to construct a 99% confidence interval for the population mean.
Step 2 :The first step is to decide whether to use the standard normal distribution or the t-distribution. The standard normal distribution is used when the population standard deviation is known, while the t-distribution is used when the population standard deviation is unknown and is estimated from the sample data.
Step 3 :In this case, we are given the sample standard deviation, not the population standard deviation. Therefore, we should use the t-distribution. This corresponds to option D: 'Use a t-distribution because it is a random sample, \(\sigma\) is unknown, and the interest rates are normally distributed.'
Step 4 :Next, we calculate the 99% confidence interval using the t-distribution. The formula for a confidence interval is: \[\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(t_{\alpha/2, n-1}\) is the t-score for a two-tailed test with \(\alpha\) level of significance and \(n-1\) degrees of freedom, \(s\) is the sample standard deviation, and \(n\) is the sample size.
Step 5 :After calculating, we get the lower and upper bounds of the 99% confidence interval.
Step 6 :The interpretation of the confidence interval is that we are 99% confident that the true population mean interest rate lies within this interval.
Step 7 :\(\boxed{\text{Final Answer: The correct distribution to use is the t-distribution (Option D). The 99% confidence interval for the population mean is approximately (3.45%, 3.93%). This means we are 99% confident that the true population mean interest rate lies within this interval.}}\)