1. The water heating experiment is described on page 338. The purpose of the experiment was to determine

the best way to heat 3 cups of water (for preparation of boxed meals) to 90°F on a kitchen stove as

quickly as possible. In this experiment, only one stove was used, and the three treatment factors were

C: diameter of pot (5.5,6.25 and 8.625 inches; coded 1, 2, 3)

D: burner size (small, large; coded 1, 2)

E: cover (no, yes; coded 1, 2).

(a) [2pts]Using the factorial form of the block-treatment model similar to (10.8.15), p. 325, but with three

treatment factors, test the hypotheses of no interactions between pairs of treatment factors, each test

done at level 0.05. Do you see any significant interactions?

(b) [2pts Test the null hypothesis that the main effects of Factor C are all equal using a = 0.05.

(c) [3pts]Find 95% simultaneous confidence intervals for all pairwise differences of the main effects of Factor

C. What do you conclude from these intervals?

(d) [3pts, optional bonus]You will see that the width of the confidence intervals in (b) is 50.3. Suppose the

experiment will be repeated in the future and what has been done will serve as a pilot study. Based

on this experiment, the 90% confidence upper limit for o² is 1200.336. Use this upper limit of o² to

calculate the necessary number of blocks for the 95% simultaneous confidence intervals for all pairwise

comparisons of the main effects of C to have a width less than or equal to 40 when the block size is

kept at 12. [Hint: If we write the response as Yhijk where h denotes block, i, j, k denote factors C, D

and E, respectively. The difference between the first two main effects is estimated by Y₁-Ŷ₂.. Think

of the variance of this estimator.]