Question

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 27 recent loans is taken. The average calculated from this sample is 5.50%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.8%. Compute 90% and 95% confidence intervals for the population mean 30-year fixed mortgage rate. Use Table 1. (Round intermediate calculations to 4 decimal places. Round "z-value" to 3 decimal places and final answers to 2 decimal places. Enter your answers as percentages, not decimals.) Confidence Level Confidence Interval 90% % to % 95% % to %

Answer 1

To compute the confidence intervals for the mean 30-year fixed mortgage rate, we use the margin of error formula and the Z-value for the desired confidence level. For a 90% confidence level, the lower and upper bounds are 5.49624% and 5.50376% respectively. For a 95% confidence level, the lower and upper bounds are 5.49560% and 5.50440% respectively.

Step-by-step explanation:

To compute the confidence intervals for the population mean 30-year fixed mortgage rate, we need to first calculate the margin of error using the formula:

Margin of Error = Z * (Standard Deviation / sqrt(Sample Size))

For a 90% confidence level, the Z-value can be obtained using Table 1 as 1.645. Plugging in the values, we get:

Margin of Error = 1.645 * (0.008 / sqrt(27)) = 0.00376

Therefore, the 90% confidence interval is calculated as:

Lower Bound = Sample Mean - Margin of Error = 5.50 - 0.00376 = 5.49624%

Upper Bound = Sample Mean + Margin of Error = 5.50 + 0.00376 = 5.50376%

Similarly, for a 95% confidence level, the Z-value can be obtained as 1.96. Plugging in the values, we get:

Margin of Error = 1.96 * (0.008 / sqrt(27)) = 0.00440

Therefore, the 95% confidence interval is calculated as:

Lower Bound = Sample Mean - Margin of Error = 5.50 - 0.00440 = 5.49560%

Upper Bound = Sample Mean + Margin of Error = 5.50 + 0.00440 = 5.50440%