As we noted, because of the regression identity, we can express the coefficient of determination in terms of the total sum of squares and the error sum of squares as $r^2=1-SSE/SST$

a. Explain why this formula shows that the coefficient of determination can also be interpreted as the percentage reduction obtained in the total squared error by using the regression equation instead of the mean. $\overline{Y}$. to predict the observed values of the response variable.

b.

$x$	$6$	$6$	$6$	$2$	$2$	$5$	$4$	$5$	$1$	$4$
$y$	$290$	$280$	$295$	$425$	$384$	$315$	$355$	$325$	$425$	$325$

What percentage reduction is obtained in the total squared error by using the regression equation instead of the mean of the observed prices to predict the observed prices?

Given In the question that,$r^2=1-SSE/SST$

we have to Explain why this formula shows that the coefficient of determination can also be interpreted as the percentage reduction obtained in the total squared error by using the regression equation instead of the mean. $\overline{Y}$. to predict the observed values of the response variable.

The coefficient of determination given by the formula:

$r^2=\frac{SSR}{SST}$

It can also be interpreted as the coefficient of variation, denoted by $r^2$;which is equal to the proportion of variation in the observed values of the response variable explained by the regression.

Now, from the concept of regression identity, total sum of squares is equal to the sum of regression sum of squares and the error sum of squares, that is,

$SSR=SST+SSE$

Therefore, the coefficient of determination can be written as

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{SST-SSE}{SST} \\ =1-\frac{SSE}{SST} \end{gathered}$$

Hence,proved.

Given in the question that,

we have to identify percentage reduction is obtained in the total squared error by using the regression equation instead of the mean of the observed prices to predict the observed prices.

The percentage reduction in the total squared error using the MINITAB can be obtained as

Step 1: Store the given data in columns named $xandy$respectively

Step 2: Choose Stat >Regression >Fitted Line Plot......

Step 3: Specify C2 in Response ($y$) and C1 in Predictor ($x$) text boxes

Step 4: Click OK.

Hence, the answer will be$r^2=93.80\%$