Question

In a random sample of 15 mortgage institutions, the mean interest rate was 3.65% and the standard deviation was 0.42%. Assume the interest rates are normally distributed. Which distribution should be used to construct the confidence interval?

A. Use a normal distribution because the interest rates are normally distributed and σ is known.
B. Use a normal distribution because n<30 and the interest rates are normally distributed.
C. Use a t-distribution because the interest rates are normally distributed and σ is known.
D. Use a t-distribution because it is a random sample, σ is unknown, and the interest rates are normally distributed.
E. Cannot use the standard normal distribution or the t-distribution because σ is unknown, n<30, and the interest rates are not normally distributed.

Answer 1

The correct distribution to construct the confidence interval for the mean interest rate is option A: Use a normal distribution because the interest rates are normally distributed, and the population standard deviation (σ) is known.

Step-by-step explanation:

When constructing a confidence interval for the mean of a normally distributed population, the choice between using a normal distribution or a t-distribution is based on whether the population standard deviation (σ) is known and the sample size (n).

Given that the interest rates are normally distributed and the population standard deviation (σ) is known (0.42%), the appropriate distribution to use for constructing the confidence interval is the standard normal distribution (option A).

The condition for using the normal distribution is satisfied because when the population standard deviation is known and the data follow a normal distribution, the standard normal distribution (Z-distribution) is used to calculate the confidence interval for the population mean. Moreover, the sample size (n = 15) is not relevant in this case when the population standard deviation is known.

Therefore, the correct choice is to use a normal distribution because the interest rates are normally distributed, and the population standard deviation (σ) is known, making option A the suitable distribution for constructing the confidence interval.