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.
A $95 \%$ prediction interval for Test4 score of a product with Test1 $= 590$ , Test2 $= 750$ , and Test $3 = 710$ is $( 583,793 )$ . Interpret this result.

A) We are $95 \%$ confident that a product's Test 4 score will fall between 583 and 793 points when the first three scores are 590,750 , and 710 , respectively.
B) Since 0 is outside the interval, there is evidence of a linear relationship between Test4 score and any of the other test scores.
C) We are $95 \%$ confident that the mean Test4 score of all manufactured products falls between 583 and 793 points.
D) We are $95 \%$ confident that a product's Test 4 score increases by an amount between 583 and 793 points for every 1 point increase in Test1 score, holding Test 2 and Test 3 score constant.