During Its Manufacture, a Product Is Subjected to Four Different $( y )$

Multiple Choice

During its manufacture, a product is subjected to four different tests in sequential order. An efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score $( y )$ , as a function of Test1 score $\left( x _ { 1 } \right)$ , Test 2 score $\left( x _ { 2 } \right)$ , and Test3 score $\left( x _ { 3 } \right)$ ) . [Note: All test scores range from 200 to 800 , with higher scores indicative of a higher quality product.] Consider the model:
$$E ( y ) = \beta _ { 1 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$$
The first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout. $\begin{array}{lrrrrr}\hline & & & & & \\\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { FVALUE } & \text { PROB >F } \\\text { MODEL } & 3 & 151417 & 50472 & 18.16 & .0075 \\\text { ERROR } & 8 & 22231 & 2779 & & \\\text { TOTAL } & 12 & 173648 & & &\end{array}$


$\begin{array}{llll}\text { ROOT MSE } & 52.72 & \text { R-SQUARE } & 0.872 \\\text { DEP MEAN } & 645.8 & \text { ADJ R-SQ } & 0.824\end{array}$


$\begin{array}{lrrrr} & \text { PARAMETER } & \text { STANDARD } & \text { T FOR 0: } & \\\text { VARIABLE } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 & \text { PROB }>|\mathrm{T}| \\\text { INTERCEPT } & 11.98 & 80.50 & 0.15 & 0.885 \\\text { X1(TEST1) } & 0.2745 & 0.1111 & 2.47 & 0.039 \\\text { X2(TEST2) } & 0.3762 & 0.0986 & 3.82 & 0.005 \\\text { X3(TEST3) } & 0.3265 & 0.0808 & 4.04 & 0.004 \\\hline\end{array}$

Suppose the $95 \%$ confidence interval for $\beta _ { 3 }$ is $( .15 , .47 )$ . Which of the following statements is incorrect?

A) We are $95 \%$ confident that the increase in Test4 score for every 1 -point increase in Test 3 score falls between .15 and .47, holding Test 1 and Test2 fixed.
B) We are $95 \%$ confident that the Test3 is a useful linear predictor of Test4 score, holding Test1 and Test2 fixed.
C) We are $95 \%$ confident that the estimated slope for the Test 4 -Test3 line falls between .15 and $.47$ holding Test1 and Test2 fixed.
D) At $\alpha = .05$ , there is insufficient evidence to reject $H _ { 0 } : \beta _ { 3 } = 0$ in favor of $H _ { \mathrm { a } } : \beta _ { 3 } \neq 0$ .