In each of Exercises 14.64-14.69, apply Procedure 14.2 on page 567 to find and interpret a confidence interval, at the specified confidence level, for the slope of the population regression line that relates the response variable to the predictor variable.
14.66 Custom Homes. Refer to Exercise 14.60; 99%.

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, $\alpha =0.01$:
$df=n-2$
$=9-2$
$=7$
According to calculation:

$t_{\alpha /2}=t_{0.01/2}$

$=t_{0.005}$

$=3.500$

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

$b_1\pm t_{\alpha /2}\frac{s_e}{\sqrt{S_{\text{II}}}}$.

Table of calculations as follows:

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

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

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

$=8316-72900/9$

$=8316-8100$

$=216$

The total sum of square SST is calculated as follows:

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

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

$=3504412-30824704/9$

$=3504412-3424967.111$

$=79444.88889$

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:

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

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

$=59.6207536$

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

$15.89351852\pm 3.500\times \frac{59.621}{\sqrt{216}}$
Also,$15.89351852\pm 14.19837459$, 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.