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

Beginner2021-12-27Added 34 answers

Step 1

Given that,

Sample size$n=25$

Sample mean$\stackrel{\u2015}{x}=0.0695$

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:

$CI=\stackrel{\u2015}{x}\pm {z}_{c}\left(\frac{\sigma}{\sqrt{n}}\right)$

$=0.0695\pm 1.645\left(\frac{0.006}{\sqrt{25}}\right)$

$=0.0695\pm 0.00197$

$=(0.068,\text{}0.071)$

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:

$CI=\stackrel{\u2015}{x}\pm {z}_{c}\left(\frac{\sigma}{\sqrt{n}}\right)$

$=0.0695\pm 2.58\left(\frac{0.006}{\sqrt{25}}\right)$

$=0.0695\pm 0.0031$

$=(0.066,\text{}0.073)$

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

Given that,

Sample size

Sample mean

Population standard deviation

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:

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:

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

lovagwb

Beginner2021-12-28Added 50 answers

We have that:

$n=25$S

$\stackrel{\u2015}{x}=6.95$

$\sigma =0.6$

To determine C.I we have formula

$\stackrel{\u2015}{x}\pm {Z}_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}$

$a+\alpha =0.10$

$l\times 0\times S$

${Z}_{\frac{\alpha}{2}}=1.645$

using equal 1

$6.95\pm \left(1.645\right)\frac{0.6}{\sqrt{25}}$

$6.95\pm 0.1974$

$(6.7526,\text{}7.1474)$

$\therefore 90\mathrm{\%}\text{}C.I(6.75,\text{}7.15)$

at $\alpha =0.01$

$l\times 0\times s$

${Z}_{\frac{\alpha}{2}}=2.576$

using equal 1

$6.95\pm \left(2.576\right)\left[\frac{0.6}{\sqrt{25}}\right]$

$6.95\pm 0.3091$

$(6.6409,\text{}7.2591)$

$\therefore 99\mathrm{\%}\text{}C.I(6.64,\text{}7.26)$

karton

Expert2022-01-04Added 613 answers

$$\begin{array}{|cc|}\hline \text{sample mean}\text{}\overline{x}& 6.950\\ \text{sample size}\text{}n=& 25.00\\ \text{std deviation}\text{}\sigma =& 0.600\\ \text{std error}\text{}=\sigma x=\frac{\sigma}{\sqrt{n}}=& 0.1200\\ \hline\end{array}$$ For 90% level: $$\begin{array}{|cc|}\hline \text{for 90\% Cl value of}\text{}z=& 1.645\\ \text{margin of error}\text{}E=z\times std\text{}error& 0.20\\ \text{lower bound=sample mean-E=}& 6.75\\ \text{Upper bound=sample mean+E=}& 7.15\\ \hline\end{array}$$ For 99% level: $$\begin{array}{|cc|}\hline \text{for 99\% Cl value of}\text{}z=& 2.576\\ \text{margin of error}\text{}E=z\times std\text{}error& 0.31\\ \text{lower bound=sample mean-E=}& 6.64\\ \text{Upper bound=sample mean+E=}& 7.26\\ \hline\end{array}$$ $$\begin{array}{|cc|}\hline \text{Confidence level}& \text{confidence interval}\\ 90\mathrm{\%}& 6.75\mathrm{\%}\text{}to\text{}7.15\mathrm{\%}\\ 99\mathrm{\%}& 6.64\mathrm{\%}\text{}to\text{}7.26\mathrm{\%}\\ \hline\end{array}$$

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