Schubert et al. ( “ The Catapult Problem: Enhanced Engineering Modeling Using Experimental Design, ” Quality Engineering , 4 , 463 – 473, 1992) conducted an experiment with a catapult to determine the effects of hook ( x1 ), arm length ( x2 ), start angle ( x3 ), and stop angle ( x4 ) on the distance that the catapult throws a ball. They threw the ball three times for each setting of the factors. The following table summarizes the experimental results.

x1 x2 x3 x4 y − 1 − 1 − 1 − 1 28.0 27.1 26.2

− 1 − 1 1 1 46.3 43.5 46.5

− 1 1 − 1 1 21.9 21.0 20.1

− 1 1 1 − 1 52.9 53.7 52.0 1 − 1 − 1 1 75.0 73.1 74.3 1 − 1 1 − 1 127.7 126.9 128.7 1 1 − 1 − 1 86.2 86.5 87.0 1 1 1 1 195.0 195.9 195.7

a. Fit a fi rst - order regression model to the data and conduct the residual analysis.

b. Use the sample variances as the basis for weighted least - squares estimation of the original data (not the sample means).

c. Fit an appropriate model to the sample variances (note: you will require an appropriate transformation!). Use this model to develop the appropriate weights and repeat part b.