1. IE241 Solutions

2. 1. The binomial probability is 3 C 3 (1/2) 3 (1/2) 0 = 1/8 Or, since the three tosses are independent, P(H on 1st, H on 2nd, H on 3rd) = P(H on 1st) x P(H on 2nd) x P(H on 3rd) = ½ x ½ x ½ = 1/8

3. 2.The binomial probability is 100 C 60 (1/2) 60 (1/2) 40

4. 3. Total Sample space: 50 C 10 Sample points in A: 40 C 10 P(A) = 40 C 10 / 50 C 10

5. 4. Sample space: 6 x 6 = 36 points (table next slide) Ways of getting RR: 4 (X boxes in table) p(RR) = 4/36 =1/9 Or since roll 1 is independent of roll 2, P(R roll1 & R roll2) = P(R roll 1) x P(R roll 2) 1/9 = 2/6 x 2/6

6. 4. Second Roll FirsTRollFirsTRoll R1R2Y1Y2B1B2 R1XX R2XX Y1 Y2 B1 B2

7. 5.(a) P(6 boys)= 6 C 6 (1/2) 6 (1/2) 0 = 1/64 and P(6 girls) = 1/64 P(6 same sex) = 1/64 + 1/64 = 1/32 (b) P(3 boys) = 6 C 3 (1/2) 3 (1/2) 3 = 20 (1/64) = 5/16

8. 6. P(4 w/1 die)) = 1/6 P(sum = 8) = 5/36 (checked boxes) Roll 2 Roll1Roll1 123456 1 2 X 3X 4X 5X 6X

9. 7. Binomial: P(3 successes; n = 5) = 5 C 3 (1/2) 3 (1/2) 2 = 10 (1/32) = 5/16

10. 8. (a) P(K|face) = P(K and face) = 4/52 = 1 P(face) 12/52 3 (b) P(bK|face) = P(bK and face)= 2/52 = 1 P(face) 12/52 6

11. 9. P(1 st = W) = 2/9 P(2 nd = B) = 3/8 P(W B) = 2/9 x 3/8 = 6/72 = 1/12 Or, sample space has 72 points, of which 6 are WB, so P(W B) = 6/72 = 1/12 W1B1 W2B1 W1B2 W2B2 W1B3 W2B3

12. 10. ♣♦♥♠ Marginal for face A1111 4/16=1/4 K1111 Q1111 J1111 Marginal for suit 4/16 =1/4

13. 11. The company can conclude that 2% of its transistors made at plant A during one particular week were defective. They can say nothing about the other plants or other weeks because their sampling frame was rstricted to one week at Plant A.

14. 12. mean = 550 s = 80 n = 100 95% confidence interval: Because the sample size is so large, t.025 = z.025 so t 025 = 1.96 and the confidence interval is or approximately, 534 ≤ μ ≤ 566

15. 13. The confidence interval based on the t distribution will always be longer than it would be if σ were known, which then would allow use of the normal distribution because a given α value for t is greater than that same α value for z unless the sample size is large.

16. 14. The length of the confidence interval is determined by and, as the sample size gets larger and larger, the estimate of the standard error gets smaller and smaller. As n → ∞ the standard error → 0.

17. 15. He can generalize his conclusions only to the 10 male students he has measured because his sample was not random.

18. 16. Mean = 4.429; Mode = 4; Median = 4.0625. To calculate the median, note that freq < 4 =19 and freq > 4 = 21. so the median lies somewhere in the range 3.5 to 4.5. Divide the frequency at 4 into two proportional distances into the range. You have 9/16 added to the lower limit and 7/16 subtracted from the upper limit. 3.5 + 9/16 = 65/16 = 4.0625 4.5 – 7/16 = 65/16 = 4.0625

19. 17. Because the distribution is completely symmetric, mean = median = mode =7. Roll 2 Roll1Roll1 123456 1 234567 2 345678 3456789 45678910 56789 11 6789101112

20. 18. Mean = 12 = np variance = 8 = npq For the binomial: So p = 1/3 variance = npq n = 36 8 = 12q q = 2/3 p = 1/3 mean = np 12 = n(1/3) n = 36

21. 19. binomial cdf point: (3,.17) P(X ≥ 4) = 1-.17 =.83

22. 20. Point on normal cdf: (1.2,.37) P(X ≥ 1.2) = 1 -.37 =.63

23. 21. Two groups s1 and n1 s2 and n2 If the two data sets are combined,

24. 22. Based on history, p = 3/30 = 1/10 P(X = 2) = 30 C 2 (.1) 2 (.9) 28 =.228 P(X = 1) = 30 C 1 (.1) 1 (.9) 29 =.141 P(X = 0) = 30 C 0 (.1) 0 (.9) 30 =.042 P(≤ 2) =.411

25. 23. p = 2/3 n = 4 (a) 4 C 4 (2/3) 4 (1/3) 0 =.198 (b) 4 C 4 (1/3) 4 (2/3) 0 =.012 So.198 +.012 =.210 (c) 6 C 3 (2/3) 3 (1/3) 3 =.219 = P(tie after 6) P(A wins 7 th ) = 2/3 P(A wins in 7) = P(tie @6) x P(A wins 7) =.219 x 2/3 =.146

26. 24. p =.1 n = 20 P(X ≥ 2) = P(at least 2) = 1- P(at most 1) 20 C 1 (.1) 1 (.9) 19 =.270 20 C 0 (.1) 0 (.9) 20 =.122 P(X ≤ 1) =.392 = P(at most 1) P(X ≥ 2) = 1 -.392 =.608

27. 25. Var(X +Y)) = Var(X) + Var(Y) + 2Cov(XY) (a) For the variance of the sum of two random variables to be greater than the sum of their variances, the variables must have a positive relationship so that their covariance > 0. (b) For the variance of the sum of two random variables to be less than the sum of their variances, the variables must have a negative relationship so that their covariance is negative.

28. 26. Mean = 400 s=15 n =16 Confidence interval

29. 27. This is a multinomial where x = 1 red ball out of 2 y = 1 green ball out of 3 z = 2 black balls out of 4 n = 4 Because of replacement, the probability for each ball remains the same.

30. 28. There are 10 ways of getting a total of 6 on 3 dice: 1 1 4 and there are 216 total 1 2 3 possibilities for 3 dice, 1 3 2 P(dice sum to 6) = 10/216 1 4 1 =.046 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 4 1 1

31. 29. It would not be surprising to find a relatively high correlation between Wall Street traffic and high tide in Maine because high tide in Maine occurs at 8 am and before that is getting higher and higher and afterward declines gradually. Also traffic in Wall Street rises from about 6 am to its peak at 8 am and then declines gradually. Of course, the correlation is meaningless.