Marenonigt

2021-12-26

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 25 recent loans is taken. The average calculated from this sample is 6.95%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.6%. Compute 90% and 99% confidence intervals for the population mean 30-year fixed mortgage rate.

yotaniwc

Step 1
Given that,
Sample size $n=25$
Sample mean $\stackrel{―}{x}=0.0695$
Population standard deviation $s=0.006$
Step 2
90% confidence intervals for the population mean 30-year fixed mortgage rate:
Critical value: The two tailed z critical value at 90% confidence level is 1.645
Calculation: The 90% confidence intervals for the population mean 30-year fixed mortgage rate can be calculated as follows:
$CI=\stackrel{―}{x}±{z}_{c}\left(\frac{\sigma }{\sqrt{n}}\right)$
$=0.0695±1.645\left(\frac{0.006}{\sqrt{25}}\right)$
$=0.0695±0.00197$

The 90% confidence intervals for the population mean 30-year fixed mortgage rate is from 6.8% to 7.1%.
Step 3
99% confidence intervals for the population mean 30-year fixed mortgage rate:
Critical value: The two tailed z critical value at 99% confidence level is 2.58.
Calculation: The 99% confidence intervals for the population mean 30-year fixed mortgage rate can be calculated as follows:
$CI=\stackrel{―}{x}±{z}_{c}\left(\frac{\sigma }{\sqrt{n}}\right)$
$=0.0695±2.58\left(\frac{0.006}{\sqrt{25}}\right)$
$=0.0695±0.0031$

The 99% confidence intervals for the population mean 30-year fixed mortgage rate is from 6.6% to 7.3%

lovagwb

We have that:
$n=25$
$\stackrel{―}{x}=6.95$
$\sigma =0.6$
To determine C.I we have formula
$\stackrel{―}{x}±{Z}_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}$
$a+\alpha =0.10$
$l×0×S$
${Z}_{\frac{\alpha }{2}}=1.645$
using equal 1
$6.95±\left(1.645\right)\frac{0.6}{\sqrt{25}}$
$6.95±0.1974$

at $\alpha =0.01$
$l×0×s$
${Z}_{\frac{\alpha }{2}}=2.576$
using equal 1
$6.95±\left(2.576\right)\left[\frac{0.6}{\sqrt{25}}\right]$
$6.95±0.3091$

karton