Question

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%.

Short answer

The increase in mean price per hundreds of square feet increase in size for the custom homes is somewhere between $\$1695$and $\$3009$at $99\%$confidence interval.

Step-by-step solution

Given information

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

The given data at Exercise 14.60 as follows:

Size	26	27	33	29	29	34	30	40	22
Price	540	555	575	577	606	661	738	804	496

Explanation

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:

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

Explanation

Table of calculations as follows:

x	y	$xy$	$x^2$	$y^2$
26	540	14040	676	291600
27	555	14985	729	308025
33	575	18975	1089	330625
29	577	16733	841	332929
29	606	17574	841	367236
34	661	22474	1156	436921
30	738	22140	900	544644
40	804	32160	1600	646416
22	496	10912	484	246016
$\sum x_i=270$	$\sum y_i=5552$	$\sum x_i{y}_i=169993$	$\sum {x_i}^2=8316$	$\sum {y_i}^2=3504412$

Explanation

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:

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

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

$=3504412-30824704/9$

$=3504412-3424967.111$

$=79444.88889$

Explanation

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:
$b_1=\frac{S_{xy}}{S_{xx}}$
$=\frac{3433}{216}$
$=15.89351852$

Explanation

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.