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. Use a normal distribution because the interest rates are normally distributed and $\sigma$ is known.
C. Use a normal distribution because $n< 30$ and the interest rates are normally distributed.
D. Use a t-distribution because the interest rates are normally distributed and $\sigma$ is known.
E. Cannot use the standard normal distribution or the t-distribution because $\sigma$ is unknown, $n< 30$, and the interest rates are not normally distributed.
Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
A. The $99 \%$ confidence interval is (Round to two decimal places as needed.).
B. Neither distribution can be used to construct the confidence interval.
Answer
Thus, the $99 \%$ confidence interval is \(\boxed{(3.17\%, 3.65\%)}\). This means that we are $99\%$ confident that the true population mean interest rate lies between $3.17\%$ and $3.65\%$.
Step 1 ：The problem provides us with a sample size of 19, a sample mean of 3.41%, and a sample standard deviation of 0.36%. We are also informed that the interest rates are normally distributed.
Step 2 ：To construct a confidence interval, we need to decide which distribution to use. 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 but the sample size is large enough (usually n > 30).
Step 3 ：In this case, we do not know the population standard deviation, but we do know the sample standard deviation and the sample size is less than 30. Therefore, we should use the t-distribution.
Step 4 ：Given that n = 19, the mean = 3.41, and the standard deviation = 0.36, we can calculate the t-score as 2.878440472713585 and the standard error as 0.08258966419340222.
Step 5 ：Using these values, we can calculate the lower bound of the confidence interval as 3.172270567957887 and the upper bound as 3.647729432042113.
Step 6 ：Thus, the $99 \%$ confidence interval is \(\boxed{(3.17\%, 3.65\%)}\). This means that we are $99\%$ confident that the true population mean interest rate lies between $3.17\%$ and $3.65\%$.
