In order to estimate the mean 30-year fixed mortgage rate for a home loan in the
ID: 3376747 • Letter: I
Question
In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 26 recent loans is taken. The average calculated from this sample is 6.60%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.5%. 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, "z" value and final answers to 2 decimal places. Enter your answers as percentages, not decimals.)
Confidence Level Confidence Interval 90% % to % 95% % to %
Explanation / Answer
For 90% confidence:
Note that
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.05
X = sample mean = 6.6
z(alpha/2) = critical z for the confidence interval = 1.64
s = sample standard deviation = 0.5
n = sample size = 26
Thus,
Lower bound = 6.439184769
Upper bound = 6.760815231
Thus, the confidence interval is
( 6.44 , 6.76 ) [answer]
****************
For 95% confidence:
Note that
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.025
X = sample mean = 6.6
z(alpha/2) = critical z for the confidence interval = 1.96
s = sample standard deviation = 0.5
n = sample size = 26
Thus,
Lower bound = 6.407806188
Upper bound = 6.792193812
Thus, the confidence interval is
( 6.41 , 6.79 ) [ANSWER]
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