Solutions for some HW problems

(Note: Some HW solutions may be distributed in paper form.)
Below are solutions for
Exercise 12

The desired p-values are given below.

(a) 0.013 (Easily obtained using StatXact. Can also get from Table D (p. 560) of G&C using r = 11.)

(b) 0.23 (Easily obtained from the table on p. 3-20 of the class notes by dividing 206 + 2 = 208 by 924 to get P(L <= 2).)

(c) 0.082 (Obtained from Table E (p. 562) of G&C, noting that r = 10.)

(d) 0.49 (Easily obtained from the table on p. 3-28 of the class notes by dividing 245249548 by 12! to get P(L >= 3), and we can subtract this from 1 to obtain P(L <= 2).)

Exercise 13

The desired p-values are given below.

(a) 0.045 (Easily obtained using StatXact.)

(b) 0.045 (Obtained using normal approximation, noting that r = 24 and a lower-tail probability is desired.)

(c) 0.36 (Obtained using normal approximation, noting that r = 38 and a lower-tail probability is desired.)

(d) 0.12 (Easily obtained from the table on p. 3-28 of the class notes, noting that l = 5 and n = 60, and we want an upper-tail probability (which can be read right off of the table).)

Exercise 14

The first observation obviously starts a new run, and for the 2nd though the (n - 1)th observations, each will start a new run if it's either the minimum or maxmimum of the set of three observations of itself and the observations immediately before and after it, which occurs with probability 2/3. Since the last observation cannot start a new run up or down, the expected number of runs is 1 + (n - 2)(2/3) = (2n - 1)/3.

Exercise 15

(a) For the value of the test statistic we have rvn = 2.965, and for the beta distribution parameter we have α = 6.369. These lead to an approximate p-value of 0.036.

(b) For the value of the test statistic we have rvn = 1.3606, and for the beta distribution parameter we have α = 30.229. These lead to an approximate p-value of 0.0055.

Exercise 16

(a) 0.370 (Easily obtained using StatXact.)

(b) 0.369 (Easily obtained using StatXact.)

(c) 0.371 (Easily obtained using StatXact.)

(d) 0.350 (Can obtain using R, Minitab, or other suitable software.)

Exercise 17

The desired p-value is 0.000, resulting from a test statistic value of 170.54.

(Note: In practice I'd write p-value < 0.0005.)

Exercise 18

Using the mean and variance to solve for the values of the two parameters, it can be detrmined that the distribution is a beta (3, 1) distribution, having cdf F(x) = x^3. Using this cdf with the method given on the top of p. 4-29 of the class notes, the p-value resulting from the K-S test is 0.00092. (Easily obtained using StatXact.)

Exercise 19

(a) 0.73 (Easily obtained using StatXact.)

(b) 0.40 (Easily obtained using StatXact.)

(Note: In 2008 I used a similar data set based on litters of 8 pigs each and strong evidence against the binomial model was obtained. But with litters of 6 pigs each we cannot reject the binomial model.)

Exercise 20

(a) 0.998 (Easily obtained using StatXact.)

(b) 0.996 (Easily obtained using StatXact.)

Exercise 21

From Table F on p. 565 of G&C we can tell that the desired value of n must exceed 40. Since Table F can't be used, one can plug into the formula 10 lines down on p. 4-32 of the class notes (putting c = 0.2 and α = 0.05) to obtain 46.111. Since this is an approximate result, 46 might work, but to play it safe one should perhaps go with 47.

Exercise 22

(a) (57, 94) (Using part of Table C on p. 553 of G&C, it can be seen that the probability that a binomial (17, 0.5) r.v. is less than or equal to 4 is about 0.0245 and the probability that a binomial (17, 0.5) r.v. is less than or equal to 12 is about 0.9755. This gives us that (X(5), X(13)) is an interval estimator for ξ0.5 which misses from below with a probability of about 0.0245, misses from above with a probability of about 0.0245, and covers with a probability of about 0.9755 - 0.0245 = 0.951.)

(b) 0.0826 (Since 4 of the observations exceed 100, using p. 5-7 of the course notes, the p-value is the probability that a binomial (15, 0.9) r.v. is less than or equal to 13.)

Exercise 23

Click here for solution.

Exercise 24

(a) 0.006 (Easily obtained using StatXact or other software (such as Minitab or R).)

(b) 0.021 (Easily obtained using StatXact or other software (such as Minitab).)

(c) 0.027 (Can obtain by a "hand calculation" (possibly using software to obtain a standard normal cdf value).)

(d) 0.011 (Easily obtained using StatXact.)

(e) 0.006 (Easily obtained using StatXact.)

(f) 0.011 (Easily obtained using Minitab, or by a "hand calculation.")

(g) 0.009 (Easily obtained using StatXact (asymptotic p-value from signed-rank test).)

(h) 0.006 (Can obtain using StatXact by doing a permutation test on the signed normal scores.)

(i) 0.012 (Can obtain using StatXact (or could use other software such as Minitab or R).)

(j) 0.006 (Easily obtained using StatXact.)

(k) 0.015 (Easily obtained using StatXact (asymptotic p-value from permutation test, or could use other software such as Minitab or R).)

Exercise 25

The desired A.R.E. is 8/9 (obtained by using the formula given on p. 5-30 of the class notes, using that the integral of the square of the pdf is 2/3 and the variance is 1/6).

Exercise 26

test p-value
(a) Student's t 0.10
(b) Welch's 0.060
(c) M-W (exact) 0.021
(d) M-W (approx w/ c.c.) 0.022
(e) M-W (approx w/o c.c.) 0.020
(f) median (exact) 0.059
(g) median (approx w/ c.c.) 0.061
(h) median (approx w/o c.c.) 0.019
(i) control median (exact) 0.12
(j) W-W (exact) 0.50
(k) W-W (approx w/ c.c.) 0.50
(l) K-S (exact) 0.033

(Note: Click here to see some information about obtaining the p-values.)

Exercise 27

test p-value
(a) Student's t 0.21
(b) M-W (exact) 0.197
(c) median (exact) 0.656
(j) W-W (exact) 0.758
(l) K-S (exact) 0.418

(Note: Click here to see some information about obtaining the p-values.)

Exercise 28

test p-value
(a) normal scores (exact) 0.019
(b) Savage scores 0.016
(c) permutation (exact) 0.013
(d) percentile modified rank (exact) 0.031
(e) Sutton scores (exact) 0.035

Exercise 29

test p-value
(a) normal scores (exact) 0.18
(b) Savage scores (exact) 0.072
(c) permutation (exact) 0.21
(d) percentile modified rank (exact) 0.21
(e) Sutton score (exact) 0.44

Exercise 30

The desired correlation is 0.919.

Exercise 31

test p-value
(a) Mood (exact) 0.082
(b) Ansari-Bradley (exact) 0.088
(c) Siegel-Tukey (exact) 0.079
(d) Klotz (exact) 0.17
(e) Conover (exact) 0.17
(f) percentile modified rank (exact) 0.30
(g) Westenberg (exact) 0.17
(h) ranklike (exact) 0.056 (Note: With a different pairing of the observations, one can get a p-value of 0.55.)

Exercise 32

The desired probability is (m+1)/NCm. (Any arrangement of the xs and ys resulting in a test statistic value of m must have k xs, followed by all n ys, followed by m - k xs, where k can be any integer from 0 to m. There are m + 1 such arrangments, and under the null hypothesis, there are NCm equally-likely possible arrangements in all.)

Exercise 33

test p-value
(a) F (approx) 0.006
(b) median (exact) 0.044
(c) Kruskal-Wallis (est. of exact) 0.010
(d) normal scores (est. of exact) 0.007
(e) percentile modified rank (est. of exact) 0.008
(f) rank analog of Tukey-Kramer (approx) 0.005 < p-value < 0.01 (based on test stat. value of 4.254)
(g) Steel-Dwass (approx) 0.01 < p-value < 0.025 (based on test stat. value of 4.040)
(h) modification of median test (exact) 0.019

Exercise 34

test p-value
(a) Jonckheere-Terpstra (exact) 0.0012
(b) Page (exact) 0.0014

(Note: Click here to see some information about obtaining the p-values.)

Exercise 35

test p-value
(a) V (exact) 0.077
(b) V* (approx) 0.037

Exercise 36

test p-value
(a) C-D (approx) 0.029
(b) F-W (exact) 0.007
(c) method from notes 0.004

Exercise 37

test p-value
(a) Kendall (exact) 2.6 × 10-5
(b) Spearman (exact) 2.8 × 10-5
(c) Pearson (exact) 1.4 × 10-7
(c) Pearson (approx) 3.3 × 10-8

Exercise 38

test p-value
(a) Mann (exact) 7.7 × 10-5
(b) Daniels (exact) 2.0 × 10-4