FE 490 Engineering Probability and Statistics Donald E.K. Martin Department of Statistics 1
Dispersion, Mean, Mode 1. The population standard deviation of the data points 2,1,6 is: (A) 1.00 (B) 1.52 (C) 2.16 (D) 2.56 (see page 33 of the manual) 2
Solution: The population variance of the data points 2,1,6 is: 1 N 2 i1 i N 1 2 2 2 14 23 13 63 3 3 21 6 ( 3) 3 x 2 The standard deviation is 14 2.16 (Answer C). 3 3
2. A certain population has a mean and standard deviation We wish to estimate the mean of this population from a sample size n taken from this population. We can increase the accuracy of our estimate by increasing: (A) (B) (C) 2 (D) n 4
We increase the accuracy of our estimate X by decreasing its variance. When we have a random sample, 1 n i i1 Var X Var X n 1 1 2 n 2 Var X 2 i 1 i n 2 n n n. Answer: D (increase n) 5
Problems 3-6 apply to the following data. Assume the data represent a population: 4,4,3,5,4,6,3 (See page 33) 6
3. The mean of the data is: (A) 4.01 (B) 4.14 (C) 4.51 (D) 4.61 4. The median of the data is: (A) 5 (B) 4 (C) 3 (D) 0 7
5. The mode of the data is: (E) 5 (F) 4 (G) 3 (H) 0 6. The variance of the data is: (A) 0.98 (B) 1.12 (C) 1.22 (D) 1.30 8
Solution to #3: 4435463 29 7 7 4.14 (Answer: B) Solution to #4: The median is the midpoint, i.e. the n 1 2 4 th observation when the data is ordered (3,3,4,4,4,5,6). So the median is 4. (Answer: B) 9
Solution to #5 The mode is the observation that occurs most frequently. Thus the mode is 4. (Answer: B) 10
Solution to #6 2 1 N N i1 x 2 N 2 2 x 2 i 1 i xi N N 2 N 2 x 2 1 i x i i 1 i N N N 2 2 N 2 x 2 1 i NN x i i1 i N i N 2 2 2 2 2 2 2 2 4 4 3 5 4 6 3 29 7 7 0.9795 (Answer: A) 2 11
Permutations and combinations (page 33) 7. An urn contains 50 balls numbered 1-50. The probability that a ball taken from this urn is even and greater than 30 is: (A) 0.10 (B) 0.15 (C) 0.20 (D) 0.25 12
Solution to #7 P ball even and > 30 X 32,34,,50 10 0.2. (Answer: C) 50 (We took a disjoint union of events, see the bottom right of page 33) 13
8. A fair coin is flipped six times. What is the probability that tails will appear at least once? (See page 33, DeMorgan s laws) (A) 0.961 (B) 0.969 (C) 0.975 (D) 0.984 14
Solution to #8 The probability of at least one tail is P at least one tail 1 P no tails 11/ 2 6 63 / 64 0.984 (Answer: D) A tail on i-th flip, Note that if i then the event in question is A A A 1 2 6, and 1 2 c 6 c c c 1 2 6 A A A A A A. 15
9. The probability that no two people in a group of six have the same birthday is most nearly: (A) 0.92 (B) 0.94 (C) 0.96 (D) 0.98 (permutations and combinations, page 33) 16
Solution to #9 The number of ways to choose six birthdays is 365 365 365. 6 (use the multiplication principle) The number of different permutations of 365 days taken 6 at a time is 17
365! P365,6 360 361 365 365 6! The answer is then 360 361 365 365 6 0.9595. (Answer: C) 18
10. The reliabilities of the first and second stages of a rocket are independent. If the reliability of the two stage rocket is required to be 0.96 and the reliability of the second stage is 0.98, what is the required reliability of the first stage? Choices: (A) 0.92, (B) 0.94, (C) 0.96, (D) 0.98 19
Solution to #10 (see page 34, top) P AB P A P B for independent A and B. P R P R R P R P R 1 2 1 2 0.96 0.96 PR 1 0.98 P R1 0.9796 0.98 where R 1- first stage works R 2- second stage works R - rocket works Answer: D 20
11. Two marksman are shooting at the same target. The probability that the first will hit the target is 0.5 and, for the second, it is 0.7. If each fires one shot, the probability that at least one will hit the target is most nearly: (A) 0.75 (B) 0.80 (C) 0.85 (D) 0.90 21
Solution to #11 PA 0.5, PB 0.7. P AB 0.5 0.7 0.5 0.7 0.85 (assume independence of shots) or c c P AB 1P A B 10.50.3 0.85. (Again, we have used DeMorgan s laws on the probability of a union, both from page 33.) 22
12. (Not in the form of an FE problem). The probability is 0.3 that a fish caught on a certain pier is a whiting. You catch five fish on this pier. What is the probability that: (A) The first three you catch are whiting 0.3 3 0.027 Answer: 23
(B) None of the five is a whiting 0.7 5 0.16807 Answer: (C) You catch exactly 3 whiting. 5 3 2 0.3 0.7 0.1323 3 Answer: (D) The first three are whiting and the others are not Answer 3 2 0.3 0.7 0.01323 24
(E) At least 3 whiting Answer: P(3)+P(4)+P(5) 5 0.1323 0.3 4 0.7 0.3 0.16308 4 1 5 (F) Fifth fish is a whiting Answer: 0.3 (independent trials) Note that we have used the binomial distribution (page 35). 25
13. The probability that exactly 3 people in a group of 8 were born on a Tuesday is most nearly: (A) 0.076 (B) 0.096 (C) 0.116 (D) 0.120 26
Solution to #13 The probability of a birth on a Tuesday is 1/7. We again have a binomial distribution (page 35). P X 81 6 3 7 7 3 5 3 3 5 1 6 56 0.0755 7 7 Answer: A 27
14. A certain population has a normal probability density function with mean 0 and variance 1. What is the probability that a single observation taken from this population is between -1 and 2? (A) 0.62 (B) 0.72 (C) 0.82 (D) 0.92 28
Solution to #14. Information on the normal probability distribution is found on page 35. We use the standard normal tables on page 42 for all problems related to the normal distribution. P 1Z 2 P 1Z 0 P 0Z 2 P 0Z 1 P 0Z 2 0.3413 0.4772 0.8185 29
Or P 1Z 2 P Z 2 P Z 1 0.9772 0.1587 0.8185. Answer: C 30
15. A certain population has a normal probability density function with mean 3 and standard deviation 2. The probability that a single observation taken from this population is less than 2 or greater than 5 is most nearly: (A) 0.37 (B) 0.41 (C) 0.45 (D) 0.49 31
Solution to #15 (see pages 33, 35, and the normal table on page 42). P X 2 P X 5 X 3 23 X 3 53 P P 2 2 2 2 P Z 0.5 P Z 1 P Z 0.5 P Z 1 0.3085 0.1587 0.4672. Answer: C 32
16. The true weight of a certain object is equally likely to be any value between 50 and 60 lbs. What is the probability that the observed weight of the object is between 52 and 52.5 lbs.? (A) 0.05 (B) 0.10 (C) 0.15 (D) 0.20 33
Solution to #16 (Uniform distribution) f x 1 1, 50 x 60; 60 50 10 0 otherwise. 52.5 1 0.5 P52 X 52.5 dx 0.05 10 10 Answer: A In general, for a Uniform(a,b) density, 52 d c Pc Xd b a length of interval total length. 34
17. A certain random variable X has a probability density function f x represented by the triangle with vertices (1,0), (5,1/4), and (9,0). What is the probability that the random variable is greater than 7? (A) 1/4 (B) 1/8 (C) 1/16 (D) 1/24 35
Solution to #17 To find the form of the density, we have that the line through (5,1/4) and (9,0) has slope -1/16 y y m x x, we and from 0 0 x, y 9,0, have, using 0 0 1 1 y 0 x9 9x f x 16 16 for 5 x 9. 36
Then P X 9 1 1 7 9x dx 16 8. 7 We could use 1 1 1 PX 7 bh 2 f(7) 2 2 8 (or similar triangles). Answer: B 37
18. A certain population has a normal probability density function with mean -2 and standard deviation 3. The probability that a single observation taken from this population is greater that 3 is most nearly: (A) 0.015 (B) 0.025 (C) 0.035 (D) 0.045 38
Solution to #18 (see normal table page 42) X PX 3 P 2 3 2 3 3 P Z 1.66 0.045. Answer: D 39
19. Linear regression (page 35) Consider the following pairs of data points: (4,2), (5,2), (6,3.5), (0.5, -2.5). From a linear least squares regression, the equation that best fits this data is: (A) y = x + 2.9 (B) y = -x + 2.9 (C) y = x - 2.9 (D) y = - x - 2.9 40
Solution to #19 Plotting the points, we have a positive slope and negative intercept. Thus the answer is C, y = x - 2.9. 41
20. Which of the following distributions would be used to determine the difference between sample means, assuming population variances are unknown? (A) 2 (B) Gaussian (C) Binomial (D) Student s t Answer: D 42