Final answer:
To construct a 99% confidence interval for the population mean, we use the t-distribution due to the small sample size (less than 30) and the fact that the population standard deviation is not known. We calculate the interval using the sample mean, standard deviation, and the appropriate t-value.
Step-by-step explanation:
To construct a 99% confidence interval for the population mean, we need to decide between using the standard normal distribution or the t-distribution. This decision depends on whether the population standard deviation is known and if the sample size is sufficiently large.
In this case, since the sample size is 24, which is less than 30, and the population standard deviation is not specified, we would typically use the t-distribution. We are assuming that the interest rates are normally distributed which allows us to use the t-distribution. To construct the 99% confidence interval, we would use the sample mean of 3.49%, the sample standard deviation of 0.39%, and the appropriate t-value for 23 degrees of freedom (since df = n - 1).
Interpreting the results of the confidence interval means that we can say with 99% confidence that the population mean of mortgage interest rates falls within the calculated interval. It does not mean that there's a 99% chance the population mean is within the interval; rather, it means that if we were to take many samples and construct a confidence interval from each of them, 99% of those intervals would contain the true population mean.