Answers for HW #1

STAT 554, Spring 2004

Print out this page, put your name on it, fill in the correct answers, and submit it at our 4th class meeting.

Problem 1

Put your answers for parts (a), (b), (c), (e), (f), (h), (k), and (l) in the answer boxes below, rounding each probability to the nearest thousandth. (For some of the parts (which are worth no points), answers are provided. These may serve as useful checks of your work.)

n rejection region size power (p = 0.6) power (p = 0.7)
30 1(a) {20, 21, ..., 30} 1(b) 0.049 1(c) 0.291 1(d) 0.730
60 1(e) {37, 38, ..., 60} 1(f) 0.046 1(g) 0.451 1(h) 0.937
120 1(i) {70, 71, ..., 120} 1(j) 0.041 1(k) 0.681 1(l) 0.998

For part (m), provide a description of the desired randomized test. (For part (n) (which is worth no points), the answer is provided. This may serve as useful check of your work.)

(m)
Reject the null hypothesis if y > 36, and reject it with probability 0.121 if y = 36.
Note: To determine the probability of rejecting for the outcome 36, one needs to solve
c P(Y = 36) + P(Y > 36) = 0.05
for the value of c, which gives us that
c = [0.05 - P(Y > 36)]/P(Y = 36),
or equivalently,
c = [P(Y <= 36) - 0.95]/P(Y = 36).
(Some students arrived at a wrong answer because of accumulated rounding error. In order to have 3 accurate significant digits for the final answer you report, you should keep more significant digits throughout the calculation, and only round to 3 significant digits at the final step.)

(n)
Reject the null hypothesis if y > 69, and reject it with probability 0.464 if y = 69.

Problem 2

Put your answers for parts (c), (d), and (e) in the answer boxes below, rounding each (approximate) p-value to the nearest thousandth. (For some of the parts (which are worth no points), answers are provided. These may serve as useful checks of your work.)

exact p-value normal approx. w/ c.c. normal approx. w/o c.c.
dealing with outcomes of red 2(a) 0.125 2(b) 0.125 2(c) 0.084
dealing with outcomes of 13 2(d) 0.050 2(e) 0.037 2(f) 0.076

Problem 3

Put your answers for parts (a) and (c) in the answer boxes below, rounding each probability to the nearest hundredth. (For part (b) (which is worth no points), the answer is provided. This may serve as useful check of your work.)

n probability
100 3(a) 0.29
200 3(b) 0.46
400 3(c) 0.65