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Math 201 Probability and Statistics 1. Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at Allergy Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure. A random sample of 1000 people with known allergies to poison ivy participated in the study. Oil from the poison ivy plant was rubbed on a patch of skin. For 500 of the subjects, it was washed off within 5 minutes. For the other 500 subjects, the oil was washed off after 5 minutes. The results are summarized in the following table. Reaction Within 5 Minutes After 5 Minutes Row Total None Mild Strong 402 51 47 60 323 117 462 374 164 Column Total 500 500 1000 Let's use the following notation for the various events: W = washing oil off within 5 minutes, A = washing oil off after 5 minutes, N = no reaction, M = mild reaction, S = strong reaction. Find the following probabilities for a person selected at random from this sample of 1000 subjects. (Use 3 decimal places.) (a) P(N) P(M) P(S) (b) P(N | W) P(S | W) (c) P(N | A) P(S | A) (d) P(N and W) P(M and W) (e) P(N or M). Are the events N = no reaction and M = mild reaction mutually exclusive? Explain. No. P(N or M) ≠ 0.Yes. P(N and M) = 0. Yes. P(N or M) = 0.No. P(N and M) ≠ 0. (f) Are the events N = no reaction and W = washing oil off within 5 minutes independent? Explain. Yes. P(N and W) ≠ P(N) · P(W).Yes. P(N and W) = P(N) · P(W). No. P(N and W) = P(N) · P(W).No. P(N and W) ≠ P(N) · P(W). 2. There is money to send two of nine city council members to a conference in Honolulu. All want to go, so they decide to choose the members to go to the conference by a random process. How many different combinations of two council members can be selected from the nine who want to go to the conference? 3. To open a combination lock, you turn the dial to the right and stop at a number; then you turn it to the left and stop at a second number. Finally, you turn the dial back to the right and stop at a third number. If you used the correct sequence of numbers, the lock opens. If the dial of the lock contains 6 numbers, 0 through 5, determine the number of different combinations possible for the lock. Note: The same number can be reused consecutively. 4. List the criteria for a binomial experiment. (Select all that apply.) The probability of success on each trial is constant.The trials are independent.The trials have exactly three outcomes.The probability of success can change from trial to trial.The trials may have two or more outcomes.The trials are mutually exclusive.A fixed number of trials repeated under identical conditions.The trials have exactly two outcomes. What does the random variable of a binomial experiment measure? The random variable measures the number of trials until the 1st success.The random variable measures the number of failures in n trials. The random variable measures the number of trials until the nth success.The random variable measures the number of successes in n trials. 5. Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. Of special interest is the study of the number of moose killed by wolves. In the period from 1958 to 1974, there were 296 moose deaths identified as wolf kills. The age distribution of the kills is as follows. Age of Moose in Years Number Killed by Wolves Calf (0.5 yr) 1-5 6-10 11-15 16-20 107 55 76 53 5 (a) For each age group, compute the probability that a moose in that age group is killed by a wolf. (Round your answers to three decimal places.) 0.5 1-5 6-10 11-15 16-20 (b) Consider all ages in a class equal to the class midpoint. Find the expected age of a moose killed by a wolf and the standard deviation of the ages. (Round your answers to two decimal places.) μ = σ = 6. A stationary store has decided to accept a large shipment of ball-point pens if an inspection of 19 randomly selected pens yields no more than two defective pens. (a) Find the probability that this shipment is accepted if 10% of the total shipment is defective. (Use 3 decimal places.) (b) Find the probability that this shipment is not accepted if 15% of the total shipment is defective. (Use 3 decimal places.) 7. According to Harper's Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 15 federal inmates is selected. (a) What is the probability that 13 or more are serving time for drug dealing? (Round your answer to three decimal places.) (b) What is the probability that 4 or fewer are serving time for drug dealing? (Round your answer to three decimal places.) (c) What is the expected number of inmates serving time for drug dealing? (Round your answer to one decimal place.) 8. Describe a normal probability distribution. (Select all that apply.) bell-shapedsymmetric around the meanmost of the data fall within 3 standard deviations from the meanbimodalmodels discrete random variablesuniform-shapedmean, median, and mode all equivalentskewed to the right 9. According to the empirical rule, approximately what percentage of the area under a normal distribution lies within 1 standard deviation? % Within 2 standard deviations? % Within 3 standard deviations? % 10. Given that x is a normal variable with mean μ = 44 and standard deviation σ = 6.2, find the following probabilities. (Round your answers to four decimal places.) (a) P(x ≤ 60) (b) P(x ≥ 50) (c) P(50 ≤ x ≤ 60) 11. Find z such that 21% of the area under the standard normal curve lies to the right of z. (Round your answer to two decimal places.) 12. The Customer Service Center in a large New York department store has determined that the amount of time spent with a customer about a complaint is normally distributed, with a mean of 9.5 minutes and a standard deviation of 2.4 minutes. What is the probability that for a randomly chosen customer with a complaint, the amount of time spent resolving the complaint will be as follows. (Round your answers to four decimal places.) (a) less than 10 minutes (b) longer than 5 minutes (c) between 8 and 15 minutes 13. Many people consider their smart phone to be essential! Communication, news, Internet, entertainment, photos, and just keeping current are all conveniently possible with a smart phone. However, the battery better be charged or the phone is useless. Battery life of course depends on the frequency, duration, and type of use. One study involving heavy use of the phones showed the mean of the battery life to be 9.75 hours with a standard deviation of 2.2 hours. Then the battery needs to be recharged. Assume the battery life between charges is normally distributed. (a) Find the probability that with heavy use, the battery life exceeds 10 hours. (Round your answer to four decimal places.) (b) You are planning your recharging schedule so that the probability your phone will die is no more than 5%. After how many hours should you plan to recharge your phone? (Round your answer to the nearest tenth of an hour.) hours 14. The University of Montana ski team has nine entrants in a men's downhill ski event. The coach would like the first, second, and third places to go to the team members. In how many ways can the nine team entrants achieve first, second, and third places? 15. Are customers more loyal in the East or in the West? The following table is based on information from a recent study. The columns represent length of customer loyalty (in years) at a primary supermarket. The rows represent regions of the United States. Less Than 1 Year 1-2 Years 3-4 Years 5-9 Years 10-14 Years 15 or More Years Row Total East 32 71 59 112 77 110 461 Midwest 31 43 68 120 63 183 508 South 53 101 93 158 106 171 682 West 41 75 67 78 45 96 402 Column Total 157 290 287 468 291 560 2053 What is the probability that a customer chosen at random has the following characteristics? (Enter your answers as fractions.) (a) has been loyal 10 to 14 years (b) has been loyal 10 to 14 years, given that he or she is from the East (c) has been loyal at least 10 years (d) has been loyal at least 10 years, given that he or she is from the West (e) is from the West, given that he or she has been loyal less than 1 year (f) is from the South, given that he or she has been loyal less than 1 year (g) has been loyal 1 or more years, given that he or she is from the East (h) has been loyal 1 or more years, given that he or she is from the West (i) Are the events from the East and loyal 15 or more years independent? Explain. Yes. These events can occur together.No. P(loyal 15 or more years) ≠ P(loyal 15 or more years | East). No. These events cannot occur together.Yes. P(loyal 15 or more years) = P(loyal 15 or more years | East). 16. Sociologists say that 80% of married women claim that their husband's mother is the biggest bone of contention in their marriages (sex and money are lower-rated areas of contention). Suppose that nine married women are having coffee together one morning. Find the following probabilities. (Round your answers to three decimal places.) (a) All of them dislike their mother-in-law. (b) None of them dislike their mother-in-law. (c) At least seven of them dislike their mother-in-law. (d) No more than six of them dislike their mother-in-law. 17. Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 7.6 minutes and a standard deviation of 2.3minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.) (a) the response time is between 3 and 9 minutes (b) the response time is less than 3 minutes (c) the response time is more than 9 minutes Mat h 201 Pro b ability and St atistics 1. Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at Allergy Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure. A random sample of 1000 people with know n allergies to poison ivy participated in the study. Oil from the poison ivy plant was rubbed on a patch of skin. For 500 of the subjects, it was washed off within 5 minutes. For the other 500 subjects, the oil was washed off