Question

To find and interpret a confidence interval , at the specified confidence level $99\%$for the slope of the population regression line that relates the response variables to the predictor variable.

Short answer

It is $99\%$sure that the rise in mean price per hundreds of square feet increases in size for custom homes will be between $1695$and$3009$.

Step-by-step solution

Given information

The given data is

Explanation

Regression $t-$interval

STEP 1: For a $99\%$confidence interval,

$\alpha =0.01$

$n=9$

$df=n-2$

$$\begin{gathered} =9-2 \\ =7 \end{gathered}$$

From the technology

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

$t_{0.005}=3.500$

The end points of the confidence interval for ${\beta }_1$are calculated using the following formula

$b_{\mathrm{I}}\pm t_{a/2}\frac{s_e}{\sqrt{S_{xx}}}$

Computation table

$S_{xy}=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n$

$=169993-\left(270\right)\left(5552\right)/9$

$=169993-1499040/9$

$$\begin{gathered} =169993-166560 \\ =3433 \end{gathered}$$

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

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

$=8316-72900/9$

$$\begin{gathered} =8316-8100 \\ =216 \end{gathered}$$

The total sum of square SST is

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

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

$$\begin{gathered} =3504412-30824704/9 \\ =3504412-3424967.111 \\ =79444.88889 \end{gathered}$$

The regression sum of squares SSR is

$SSR=\frac{S_{xy}^2}{S_{xx}}$

$=\frac{(3433{)}^2}{216}$

$=\frac{11785489}{216}$

$SSE=SST-SSR$

$$\begin{gathered} =79444.88889-54562.44907 \\ =24882.43981 \end{gathered}$$

The slope of the regression line is calculated using the formula,

$b_1=\frac{S_{xy}}{S_{xx}}$

$=\frac{3433}{216}$

$=15.89351852$

The formula for calculating the standard error of the estimate is,

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

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

localid="1652459077820" $$\begin{gathered} =59.6207536 \\ \approx 59.621 \end{gathered}$$

Substituting the values

$15.89351852\pm 3.500\times \frac{59.621}{\sqrt{216}}$

$15.89351852\pm 14.19837459$or localid="1652459180822" $1.695to30.091$

We can be $99\%$sure that the rise in mean price per hundreds of square feet increase in size for custom homes will be between $1695$and $3009$.