STAT2910_Chapter 4&5_Tutorial_W24

Department of Mathematics and Statistics STAT2910-01: Statistics for Sciences Faculty of Science University of Windsor Tutorial for Chapter 4 and 5 Questions for Chapter 4 Exercise 1. Of the delegates at a convention, 60% attended the breakfast forum, 70%attended the dinner speech, and 40% attended both events. Define events A and B as follows: A: attended the breakfast forum B: attended the dinner speech What is the probability that a randomly selected delegate either attended the breakfast forum, or attended the dinner speech, or attended both? Exercise 2. Heidi prepares for an exam by studying a list of 15 problems. She can solve 9 of them. For the exam, the instructor selects 7 questions at random from the list of 15. What is the probability that Heidi can solve all 7 problems on the exam? Exercise 3. a. How many different combinations of 5 students can be drawn from a class of 25 students? b. How many permutations of 3 colours can be drawn from a group of 20 colours? Exercise 4. A psychologist tests Grade 7 students on basic word association skills and number pattern recognition skills. Let W be the event a student does well on the word association test. Let N be the event a student does well on the number pattern recognition test. A student is selected at random, and the following probabilities are given: P ( W ∩ N ) = 0 . 25 , P W ∩ N ∁ = 0 . 15 , P W ∁ ∩ N = 0 . 10 , P W ∁ ∩ N ∁ = 0 . 50 a. What is the probability that the randomly selected student does well on the word association test? b. What is the probability that the randomly selected student does well on the number pattern recognition test? c. What is the probability that the randomly selected student does well on at least one of the tests? d. If the randomly selected student does well on the word association test, what is the probability he or she will also do well on the number pattern recognition test? e. If the randomly selected student does well on the number pattern recognition test, what is the proba- bility he or she will also do well on the word association test? f. Are the events W and N mutually exclusive? Justify your answer. g. Are the events W and N independent? Explain. 1

Exercise 5. A researcher studied the relationship between the salary of a working woman with school-aged children and the number of children she had. The results are shown in the following probability table: Salary 2 or fewer Children More than 2 Children High salary 0.13 0.2 Medium salary 0.20 0.10 Low salary 0.30 0.25 Let A denote the event that a working woman has two or fewer children, and let B denote the event that a working woman has a low salary. a. What is the probability that a working woman has two or fewer children? b. What is the probability that a working woman has a low salary? c. What is the probability that a working woman has two or fewer children and has a low salary? d. What is the probability that a working woman either has two or fewer children or has a low salary? e. If a working woman has two or fewer children, what is the probability that she has a low salary? f. If a working woman has a low salary, what is the probability that she has two or fewer children? g. From this information, can one conclude that the salary of a working woman with school-aged children and the number of children she has are independent events? Explain. Exercise 6. A federal agency is trying to decide which of two waste management projects to investigate as the source of air pollution. In the past, projects of the first type were in violation of air quality standards with probability 0.3 on any given day, while projects of the second type were in violation of air quality standards with probability 0.25 on any given day. It is not possible for both projects to pollute the air in one day. Let A i , i = 1 , 2, denote that project of type i was in violation of air quality standards. a. Find the probability of an air pollution problem being caused by either the first project or the second project. b. If the first project is violating air quality standards, what is the probability the second project is also violating federal air quality standards? Exercise 7. An experiment was conducted in which rats could choose to enter one of two corridors, A or B. A random sample of three rats is selected. Let X = number of rats that select corridor B. a. Assuming the rats select their favourite corridor independently of one another and that the two corri- dors are equally likely to be selected, find the probability distribution of X . b. What is the probability that, at most, one rat selects corridor B? c. What is the probability that at least one rat selects corridor B? 2

Exercise 11. An oil firm plans to drill 20 wells, each having a probability 0.2 of striking oil. Each well costs $20 , 000to drill; a well that strikes oil will bring in $750 , 000 in revenue. Find the expected gain from the 20 wells. Exercise 12. From past experience, it is known 90% of one-year-old children can distinguish their mother’s voice from the voice of a similar-sounding female. A random sample of 20 one-year-olds are given this voice recognition test. a. Find the probability that at least three children do not recognize their mother’s voice. b. Find the probability that all 20 children recognize their mother’s voice. c. Let the random variable X denote the number of children who do not recognize their mother’s voice. Find the mean of X . d. Let the random variable X denote the number of children who do not recognize their mother’s voice. Find the variance of X . e. Find the probability that, at most, four children do not recognize their mother’s voice. Exercise 13. Hotels, like airlines, often overbook, counting on the fact that some people with reservations will cancel at the last minute. A certain hotel chain has found that 20% of the reservations will not be used. a. If we randomly selected 15 reservations, what is the probability more than 8 but fewer than 12 reser- vations will be used? b. If four reservations are made, what is the chance fewer than two will cancel? Exercise 14. Let X be a binomial random variable with n = 15 and p = 0 . 20. a. Calculate P ( X ≤ 4) using the binomial formula. b. Calculate P ( X ≤ 4) using the cumulative binomial probabilities table in Appendix I. c. Compare the results of (a) and (b). d. Calculate the mean and standard deviation of the random variable X . e. Use the results of (d) to calculate the intervals µ ± σ , µ ± 2 σ and µ ± 3 σ . Find the probability that an observation will fall into each of these intervals. f. Are the results of part (e) consistent with Tchebysheff’s Theorem? With the Empirical Rule? Exercise 15. The number of telephone calls coming into a business’s switchboard averages four calls per minute. Let X be the number of calls received. a. Find P ( X = 0). b. What is the probability there will be at least one call in a given one-minute period? c. What is the probability at least one call will be received in a given two-minute period? 4

Exercise 16. The quality of computer disks is measured by sending the disks through a certifier that counts the number of missing pulses. A certain brand of computer disks averages 0.1 missing pulses per disk. Let the random variable X denote the number of missing pulses. a. What is the distribution of X ? b. Find the probability that the next inspected disk will have no missing pulse. c. Find the probability the next disk inspected will have more than one missing pulse. d. Find the probability neither of the next two disks inspected will contain any missing pulse. Exercise 17. A shipment of six parrots from Brazil includes two parrots with a potentially fatal disease. As usual, the Customs Office at the shipment’s point of entry randomly samples two parrots and tests them for disease. Let the random variable X be the number of healthy parrots in the sample. Find the probability distribution of X . Exercise 18. An eight-cylinder automobile engine has two misfiring spark plugs. The mechanic removes all four plugs from one side of the engine. a. What is the probability the two misfiring spark plugs are among those removed? b. What is the mean number of misfiring spark plugs? c. What is the variance of the number of misfiring spark plugs? 5