Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning Statistics for Engineers Midterm Review 1. In a lot of 10 components, 2 are sampled at random for inspection. Assume that in fact exactly 2 of the 10 components in the lot are defective. Let X be the number of sampled components that are defective. a) Find the probability mass function of X. b) Find the mean of X. (Ans: 0.4) c) Find the standard deviation of X. (Ans: 0.533) d) Find the proportion of at most one component being defective. (Ans: 0.978) 2. Let A and B be events with P(A)=0.3 and P(AUB)=0.7. a) For what value of P(B) will A and B be mutually exclusive?(Ans: 0.4) b) For what value of P(B) will A and B be independent?(Ans: 4/7) Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 3. Suppose A, B, and C are independent events, and that P(A)=1/3, P(B)=2/5, and P(C)=3/4. a) What is P( A C ) ? Ans: 5/6 b) What is P( A B) ? Ans: 1/5 c) What is P( A ( B C ) ) ? Ans: 8/15 4. Find the values of a and b for the given pdf if the expected value is 3/5. (Ans:𝑎 = 3/5, 𝑏 = 6/5) 2 0≤𝑥≤1 𝑓(𝑥) = { 𝑎 + 𝑏𝑥 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 5. The strength observations for certain bars is given by the following stem and leaf plot. Stem 1 2 3 4 5 6 7 Leaf Unit = 10 Leaf 19 59 456689 4789 0 3457 23 Which of the following statements are true about this stem and leaf plot? a) b) c) d) The smallest value in the data set is 11. If redone with 2 leaf categories per stem the increment would be 5. One of the data points is 50. The median is 440 and mode is 360. Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 6. Do the summary statistics for A provide enough information to construct a boxplot? 7. A machine produces parts that are acceptable if they are larger than 10.5 mm and smaller than 10.6 mm. From past tests, it is known that the parts produced follow a normal distribution with a mean of 10.56 mm and variance of 0.0004 mm2 . a) What proportion parts fall within the tolerance? (Ans:0.9759) b) Out of a batch of 400 parts, how many would be acceptable? (Ans: 390) c) A certain part is 0.5 standard deviations above the mean. What proportion of parts produced are larger than that? (Ans: 0.3085) Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 8. The lifetime of a bearing (in years) follows the Weibull distribution with parameters = 1.5 and = 0.8 . a) What is the probability that a bearing lasts more than 1 year? (Ans: 0.4889) b) What is the probability that a bearing lasts less than 2 years? (Ans: 0.8679) 9. The lifetime (in days) of a certain electronic component is lognormally distributed with = 3.5 and = 1.2 . a) Find the mean lifetime. (Ans: 68.0) b) Find the probability that a component lasts between 50 and 250 days. (Ans:0.3204) c) Find the median lifetime. (Ans: 33.1) 10. A commuter must pass through three traffic lights on her way to work. For each light, the probability that it is green when she arrives is 0.8. The lights are independent. The commuter goes to work five days per week. Let X be the number of times out of the five days in a given week that all three lights are green. Assume the days are independent of one another. What is the distribution of X? a) b) c) d) X ∼ Bin(5, 0.8) X ∼ Bin(5, 0.512) X ∼ Bin(3, 0.8) X ∼ Bin(3, 0.512) Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 11. Assume scores on an entrance test follows a normal distribution with mean of 64 points and standard deviation of 2.5. What is the 90th percentile? a) 64.8 b) 66.5 c) 67.2 d) 90.0 12. The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. What is the probability that the sample mean breaking strength for a random sample of 36 rivets is between 9900 and 10,200? (Ans:0.8767) Check out this web app to visualize a sampling distribution to understand CLT. 13. Aluminum cans received by a beverage company are tested for conformance to a strength specification. From a very large lot of cans, 100 are sampled at random and tested. The shipment is rejected if 12 or more of the cans fail the test. Assume that 20% of the cans in the lot do not meet the specification. What is the probability that the lot will be rejected? (Ans: 0.9834) Check out this web app to visualize the binomial distribution for large n. Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 14. It is known that cars arrive at a drive-through at a rate of three cars per minute between 12 noon and 1:00 p.m. Assuming the number of cars that arrive in any time interval has a Poisson distribution, what is the probability that exactly 9 cars arrive between 12:10 and 12:15? (Ans: 0.0324) 15. Jones battery company makes 12-volt batteries. After many years of product testing, Jones knows the average life of a battery is normally distributed, with a mean age of 45 months, and a standard deviation of 7 months. a) Jones guarantees to replace for free any battery that fails in less than 37 months. What percentage of the total production does Jones expect to replace? (Ans: 12.71%) b) If Jones wishes to replace no more than 3.75% of their production, how long should the battery be guaranteed? (Ans: 32 months) c) Find the probability that a randomly selected battery will have a lifetime over 37 months given that it has already lasted 30 months. (Ans: 88.7%) Student Learning Centre studentlearning@ontariotechu.ca ontariotechu.ca/studentlearning 16. The number of telephone calls that pass through a switchboard has a Poisson distribution with mean equal to 2 per minute. a) Find the expected number of phone calls that pass through the switchboard in one minute. b) Find the probability that no telephone calls pass through the switchboard in two consecutive minutes. (Ans: 0.0183) 17. An exam has two questions: I and II. The probability that a student will solve only question I correctly is 0.4, and only question II is 0.45. The probability that a student will solve either question I or II correctly is 0.9. Find the probability that the student will correctly solve: a) Both questions (Ans: 0.05) b) Exactly one question (Ans: 0.85) c) None of the questions (Ans: 0.1) d) Question II given that he solved question I correctly (Ans: 0.11) 3 18. Find the missing lower bound so 𝑓(𝑡) is a pdf on [a, 4]. (Ans: 𝑎 = √63) 4 ∫ 3𝑡 2 𝑑𝑡 𝑎