Question

14.94 Custom Homes. Following are the size and price data for custom homes from Exercise 14.24.

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

a. Determine a point estimate for the mean price of all $2800-sq.ft.$Equestrian Estate homes.
b. Find a $99\%$confidence interval for the mean price of al $2800-sq.ft.$Equestrian Estate homes.
c. Find the predicted price of a $2800-sq.ft.$Equestrian Estate home
d. Determine a $99\%$prediction interval for the price of a $2800-sq.ft$ Equestrian Estate home.

Short answer

(a) A point estimate for the mean price is ${\hat{y}}_p=585.10$.

(b) A $99\%$confidence interval for the mean price is between $\$510$to $\$660$.

(c) The predicted price is${\hat{y}}_p=585.10$.

(d) A $99\%$confident that the price is between $\$363$to $\$807$.

Step-by-step solution

Part (a) Step 1: Given information

To determine a point estimate for the mean price of all$2800-sq.ft.$ Equestrian Estate homes.

Part (a) Step 2: Explanation

Calculation of the table 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
$\sum x_i=270$	$\sum y_i=5552$	$\sum x{y}_i=169993$	$\sum {x_i}^2=8316$	$\sum {y_i}^2=3504412$

Part (a) Step 3: Explanation

Also, $S_{xy}=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n$
$=169993-\left(270\right)\left(5552\right)/9$
$=169993-1499040/10$
$=169993-166560$
$=3433$

Then,

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

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

$=8316-72900/9$

$=8316-8100$

$=216$

Part (a) Step 4: Explanation

Note that, $SST$ stands for total sum of squares.
$S_{yy}=\left(\sum y_i^2\right)-{\left(\sum y_i\right)}^2/n$
$=3504412-{\left(5552\right)}^2/9$
$=3504412-30824704/9$
$=3504412-3424967.111$
$=79444.88889$
The $SSR$ regression sum of squares is determined as:
$SSR=\frac{S_{xy}^2}{S_{xx}}$
$=\frac{{\left(3433\right)}^2}{216}$
$=\frac{11785489}{216}$
$=54562.44907$

Then, $SSE=SST-SSR$

$=79444.88889-54562.44907$

$=24882.43981$

Part (a) Step 5:Explanation

The slope of the regression line is determined as:
$b_i=\frac{S_{xy}}{S_{xx}}$
$=\frac{3433}{216}$
$=15.89351852$
The standard error is calculated as:
$s_e=\sqrt{\frac{SSE}{n-2}}$
$=\sqrt{\frac{24882.43981}{9-2}}$
$=59.6207536$
$\approx 59.621$

As a result, the regression equation is ${\hat{y}}_p=140.0838889+15.8935x_p$
By substituting the value of $x_p=28$ in the regression equation, the formula for computing the value of the point estimation is achieved as:
${\hat{y}}_p=140.0838889+15.8935x_p$
$=140.0838889-15.8935\left(28\right)$
$=585.101889$
As a result, the point estimate is ${\hat{y}}_p=585.10$.

Part (b) Step 1: Given information

To find $99\%$confidence interval for the mean price of all $2800$ -sq.ft. Equestrian Estate homes.

Part (b) Step 2: Explanation

Let, $a=0.01$for a $99\%$ confidence interval.
Since $n=9$,
$df=n-2$
$=9-2$
$=7$
From calculation, $t_{a/2}=t_{0.01/2}=t_{0.005}=3.500$

Computing the confidence interval end points for the conditional mean of the response variable is as follows:

${\hat{y}}_p\pm t_{a/2}{s}_e\sqrt{\frac{1}{n}+\frac{{\left(x_p-\sum x_i/n\right)}^2}{S_{xi}}}$

Note that, $x_p=28$

${\hat{y}}_p=585.10$

$S_e=59.621$

$S_{xx}=216$

Hence, $585.10\pm 3.500(59.621)\sqrt{\frac{1}{9}+\frac{(28-270/9{)}^2}{216}}$

Or,

$585.10\pm 75.1310469,$or $509.97$to $660.231$

As a result, $99\%$confident that the mean price of all $2800$-sq-ft Equestrian Estate homes is somewhere between $\$510$to$\$660$.

Part (c) Step 1: Given information

To find the predicted price of a$2800$ sq.ft. Equestrian Estate home.

Part (c) Step 2: Explanation

Let, the regression equation is ${\hat{y}}_p=140.0838889+158935x_p$
By substituting the value of $x_p=28$ in the regression equation, the projected value is obtained as follows:
${\hat{y}}_p=140.0838889+158935x_p$
$=140.0838889-15.8935\left(28\right)$
$=585.101889$
As a result, the predicted value is ${\hat{y}}_p=585.10$

Part (d) Step 1: Given information

To determine a $99\%$prediction interval for the price of a $2800$sq.ft. Equestrian Estate home.

Part (d) Step 2: Explanation

Let, $a=0.01$for a $99\%$confidence interval.

Since, $n=9$

$df=n-2$

$=9-7$

$df=2$

From calculation, $t_{a/2}=t_{0.01/2}=t_{0.005}=3.500$

Computing the confidence interval end points for the conditional mean of the response variable is as follows:
${\hat{y}}_p\pm t_{a/2}{s}_e\sqrt{\frac{1}{n}+\frac{{\left(x_p-\sum x_i/n\right)}^2}{S_{xx}}}$
Note that, $x_p=2800$
${\hat{y}}_p=44641.9$
$S_e=59.621$
$S_{xx}=216$
Hence,

$585.10\pm 3.500(59.621)\sqrt{1+\frac{1}{9}+\frac{(28-270/9{)}^2}{216}}$

Also,

$585.10\pm 221.7866177$, or $363.31$to $806.89$

As a result, $99\%$confident that the price of a$2800$-sq-ft Equestrian Estate home is somewhere between $\$363$to $\$807$.