91Ó°ÊÓ

To find the confidence interval at $99\%$that refer to Exercise 14.60.

The given data at Exercise 14.60 as follows:

The regression $t$-interval as follows:
Since $n=9$for a $99\%$confidence interval, $\mathrm{Î\pm }=0.01$:
$df=n-2$
$=9-2$
$=7$
According to calculation:

$t_{\mathrm{Î\pm }/2}=t_{0.01/2}$

$=t_{0.005}$

$=3.500$

For ${\mathrm{β}}_1$, the formula for finding the end points of the confidence interval is as follows:

$b_1Â\pm t_{\mathrm{Î\pm }/2}\frac{s_e}{\sqrt{S_{\text{II}}}}$.

Table of calculations as follows:

Let,$S_{xy}=∑x_i{y}_i-\left(∑x_i\right)\left(∑y_i\right)/n$
$=169993-\left(270\right)\left(5552\right)/9$
$=169993-1499040/9$
$=169993-166560$
$=3433$

Then, $S_{xx}=∑x_i^2-{\left(∑x_i\right)}^2/n$

$=8316-(270{)}^2/9$

$=8316-72900/9$

$=8316-8100$

$=216$

The total sum of square SST is calculated as follows:

$S_{yy}=∑y_i^2-{\left(∑y_i\right)}^2/n$

$=3504412-(5552{)}^2/9$

$=3504412-30824704/9$

$=3504412-3424967.111$

$=79444.88889$

The regression sum of squares $SSR$ is calculated as follows:
$SSR=\frac{{S_{xy}}^2}{S_{xx}}$
$=\frac{(3433{)}^2}{216}$
$=\frac{11785489}{216}$
$=54562.44907$
$\text{SSE}=SST-SSR$
$=79444.88889-54562.44907$
$=24882.43981$
The slope of the regression line can be calculated using the following formula:
$b_1=\frac{S_{xy}}{S_{xx}}$
$=\frac{3433}{216}$
$=15.89351852$

The standard error of the estimate is calculated using the formula:

$s_e=\sqrt{\frac{SSE}{n-2}}$

$=\sqrt{\frac{24882.43981}{9-2}}$

$=59.6207536$

$≈59.621$
Let, $b_1=15.89351852$
$s_e=59.621$
$S_{xx}=216$
Hence,

$15.89351852Â\pm 3.500×\frac{59.621}{\sqrt{216}}$
Also,$15.89351852Â\pm 14.19837459$, Or $1.695$to $30.091$
As a result, the increase in mean price per hundreds of square feet increase in size for the custom homes is somewhere between $\$1695$and $\$3009$at $99\%$confident.