Humble Independent School District
Chapter 6 Review packetOther1.The probability distribution below is for the random variable X = number of mice caught in traps during a single night in small apartment building.X012345P(X)0.120.200.310.140.160.07(a) Make a histogram of this probability distribution in the grid:(b) Describe in words and find its value.(c) Express the event “trapping at least one mouse” in terms of X and find its probability.2.Mr. Voss and Mr. Cull bowl every Tuesday night. Over the past few years, Mr. Voss’s scores have been approximately Normally distributed with a mean of 212 and a standard deviation of 31. During the same period, Mr. Cull’s scores have also been approximately Normally distributed with a mean of 230 and a standard deviation of 40. Assuming their scores are independent, what is the probability that Mr. Voss scores higher than Mr. Cull on a randomly-selected Tuesday night?3.Joe the barber charges $32 for a shave and haircut and $20 for just a haircut. Based on experience, he determines that the probability that a randomly selected customer comes in for a shave and haircut is 0.85, the rest of his customers come in for just a haircut. Let J = what Joe charges a randomly-selected customer.(a) Give the probability distribution for J.(b) Find and interpret the mean of J, .(c) Find and interpret the standard deviation of J, .4.The probability distribution below is for the random variable X = number of medical tests performed on a randomly selected outpatient at a certain hospital.X012345P(X)0.330.200.180.140.120.03(a) Make a histogram of this probability distribution in the grid:(b) Describe in words and find its value.(c) Express the event “performing at least two tests” in terms of X and find its probability.5.The mean height of players in the National Basketball Association is about 79 inches and the standard deviation is 3.5 inches. Assume the distribution of heights is approximately Normal. Let H = the height of a randomly-selected NBA player. Find and interpret .6.Man Hong is running the balloon darts game at the school fair. He has blown up hundreds of balloons with notes about prize tickets inside them. Twelve percent of the notes say “You win 5 tickets,” twenty percent say “You win 3 tickets,” and the rest say “Sorry, try again!’ After each play, he replaces the popped balloon with another one bearing the same note. Let T = the number of tickets won by a randomly selected player of this game.(a) Give the probability distribution for T.(b) Find and interpret the mean of T, .(c) Find and interpret the standard deviation of T, .7.A four sided die shaped like an asymmetrical tetrahedron has the following roll probabilities.Number on Die1234Probability0.40.30.20.1Let X = the result of a single roll.(a) Find (b) Find (c) Describe in words and find its value.(d) Find the smallest value A for which (e) If T = the sum of two rolls, find (f) Find and interpret the mean and standard deviation of X.8.As of December 2008, various polls indicate that 35% of people who use the internet have profiles on at least one social networking site. We will discover later in the text that if you take a sample of 50 internet users, the proportion of the sample who have profiles on social networking sites can be considered a random variable. Moreover, assuming the 35% is accurate, this random variable will be approximately Normally distributed with a mean of 0.35 and a standard deviation of 0.067. What is the probability that the proportion of a sample of size 50 who have profiles on social networking sites is greater than 0.5?9.A game show host has developed a new game called “Grab All You Can." Here’s how it works: a contestant reaches his dominant hand (i.e. right hand for right-handed people) into a jar of $10 bills and grabs as many as he can in one handful. Then he does the same thing with his non-dominant hand in a jar of $20 bills. Research with many volunteers has determined that the mean number of $10 bills drawn is 68 with a standard deviation of 9.5, and the mean number of $20 dollar bills is 58, with a standard deviation of 7.8. (a) If D = the amount of money, in dollars, that a randomly-selected contestant grabs from the “$10 grab,” find the mean and standard deviation of D.(b) If T = the total amount of money, in dollars, that a contestant grabs from both jars, find the mean and standard deviation of T.(c) The game’s rules are that a contestant must pay $500 (from previous winnings) for one round of play (that is, one grab from each jar). If G = how much the contestant gains from one round of play, find the mean and standard deviation of G.10.Suppose that the mean height of policemen is 70 inches with a standard deviation of 3 inches. And suppose that the mean height for policewomen is 65 inches with a standard deviation of 2.5 inches. If heights of policemen and policewomen are Normally distributed, find the probability that a randomly selected policewoman is taller than a randomly selected policeman.11.The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do. (a) A single toss of a balanced coin results in either 0 or 1 head, each with probability 1/2. What are the mean and standard deviation of the number of heads? (b) Toss a coin four times. Use the rules for means and variances to find the mean and standard deviation of the total number of heads.12.You have two instruments with which to measure the height of a tower. If the true height is 100 meters, measurements with the first instrument vary with mean 100 meters and standard deviation 1.2 meters. Measurements with the second instrument vary with mean 100 meters and standard deviation 0.65 meters. You make one measurement with each instrument. Your results are X1 for the first and X2 for the second and the measurements are independent.(a) To combine the two measurements, you might average them, . What are the mean and standard deviation of Y? (b) It makes sense to give more weight to the less variable measurement because it is more likely to be closer to the truth. Statistical theory says that to make the standard deviation as small as possible you should weight the two measurements inversely proportional to their variances. The variance of X2 is very close to half the variance of X1, so X2 should get twice the weight of X1. That is, use What are the mean and standard deviation of W?13.Lamar and Lawrence run a two-person lawn-care service. They have been caring for Mr. Johnson’s very large lawn for several years, and they have found that the time it takes Lamar to mow the lawn itself is approximately Normally distributed with a mean of 105 minutes and a standard deviation of 10 minutes. Meanwhile, the time it takes for Lawrence to use the edger and string trimmer to attend to details is also Normally distributed with a mean of 98 minutes and a standard deviation of 15 minutes. They prefer to finish their jobs within 5 minutes of each other. What is the probability that this happens, assuming their finish times are independent?14.Determine whether each random variable described below satisfies the conditions for a binomial setting, a geometric setting, or neither. Support your conclusion in each case.(a) Suppose that one of every 100 people in a large community is infected with HIV. You want to identify an HIV-positive person to include in a study of an experimental new drug. How many individuals would you expect to have to interview in order to find the first person who is HIV-positive?(b) Deal seven cards from a standard deck of 52 cards. Let H = the number of hearts dealt.15.Determine whether each random variable described below satisfies the conditions for a binomial setting, a geometric setting, or neither. Support your conclusion in each case.(a) A high school principal goes to 10 different classrooms and randomly selects one student from each class. X = the number of female students in his group of 10 students.(b) You are on Interstate 80 in Pennsylvania, counting the occupants in every fifth car you pass. Let Z = the number of cars you pass before you see one with more than two occupants.16.Suppose that 20% of a herd of cows is infected with a particular disease.(a) What is the probability that the first diseased cow is the 3rd cow tested?(b) What is the probability that 4 or more cows would need to be tested until a diseased cow was found?17.A manufacturer produces a large number of toasters. From past experience, the manufacturer knows that approximately 2% are defective. In a quality control procedure, we randomly select 20 toasters for testing. (a) Determine the probability that exactly one of the toasters is defective.(b) Find the probability that at most two of the toasters are defective. (c) Let X = the number of defective toasters in the sample of 20. Find the mean and standard deviation of X.18.Determine whether each random variable described below satisfies the conditions for a binomial setting, a geometric setting, or neither. Support your conclusion in each case.(a) Draw a card from a standard deck of 52 playing cards, observe the card, return the card to the deck, and shuffle. Count the number of times you draw a card in this manner until you observe a jack. (b) Joey buys a Virginia lottery ticket every week. X is the number of times in a year that he wins a prize.19.When a computerized generator is used to generate random digits, the probability that any particular digit in the set {0, 1, 2, . . . , 9} is generated on any individual trial is 1/10 = 0.1. Suppose that we are generating digits one at a time and are interested in tracking occurrences of the digit 0.(a) Determine the probability that the first 0 occurs as the fifth random digit generated.(b).How many random digits would you expect to have to generate in order to observe the first 0?(c)Let X = number of digits selected until first zero is encountered. Construct a probability distribution histogram for X = 1 through X = 5. Use the grid provided.20.Suppose there are 1100 students in your high school, and 28% of them take Spanish. You select a sample of 50 student in the school, and you want to calculate the probability that 15 or more of the students in your sample take Spanish. Which condition for the binomial setting has been violated here, and why does the binomial distribution do a good job of estimating this probability anyway?21.A fair coin is flipped 20 times. (a) Determine the probability that the coin comes up tails exactly 15 times.(b).Let X = the number of tails in the 20 flips. Find the mean and standard deviation of X.(c) Find the probability that X takes a value within 1 standard deviation of its mean.Chapter 6 Review packetAnswer SectionOTHER1.ANS:A. See histogram at right. B. The probability that two or more mice are caught during a single night; 0.68. C. .PTS:12.ANS:Let D = difference in scores between Mr. Cull and Mr. Voss. Then PTS:13.ANS:A. B. = the mean amount of money Joe can expect to make per customer in the long run = C. = the “typical” distance from the mean ($30.20) for each individual customer = PTS:14.ANS:A. See histogram at right. B. The probability that an outpatient undergoes no more than 3 medical tests; 0.85. C. .PTS:15.ANS:This is the probability that a randomly-selected NBA player’s height is greater than 74 inches.PTS:16.ANS:A. B. = the mean number of tickets won per contestant in the long run = 0(0.68)+3(0.2)+5(0.12)=1.2 ticketsC. = the “typical” distance from the mean (1.2) for each individual contestant = ticketsPTS:17.ANS:A. . B. C. The probability of rolling a 3, given that the roll is 2 or greater; . D. so A = 3. E. The event T = 4 can happen three ways:{1, 3}, {3, 1}, and {2, 2}. Probabilities for these events are, respectively, . Hence the total probability is 0.25. F. ; . is the expected mean roll if the die is rolled many times, or the expected long-run value of a single roll. is the typical distance each roll is from the mean roll.PTS:18.ANS:PTS:19.ANS:PTS:110.ANS:Let D = difference in height between random male and randomfemale. Then PTS:111.ANS:PTS:112.ANS:PTS:113.ANS:3. We are interested in the difference between their job completion times, so minutes ;. To be within 5 minutes means ±5 minutes. Therefore PTS:114.ANS:A. This is a geometric setting: we are looking for the number of independent trials it takes to get the first person who is HIV positive. B. This is neither binomial nor geometric, because each trial is not independent of previous trials since it is done without replacement: the suit of the first card influences the probability of a heart on the second card, and so on.PTS:115.ANS:A. This is neither binomial or geometric, because the probability of success (selecting a female student) is probably different in each trial, unless every classroom has exactly the same proportion of females, which is unlikely. B. This is a geometric setting: we are counting the number of independent trials (cars) it takes to get the first car with more than two occupants.PTS:116.ANS:a. b. PTS:117.ANS:PTS:118.ANS:A. This is a geometric setting: We are counting the number of cards it takes to get our first jack. (Since we are selecting a card with replacement, the trials are independent). B. This is a binomial setting: Binary outcomes (Joey wins or loses), Independent trials (whether he wins this week does not influence whether he wins next week), Number of trials is fixed at 52 (weeks), and the probability of Success—however minuscule—does not change.PTS:119.ANS:A. Geometric probability: B. Geometric mean; . C. Histogram below PTS:120.ANS:The condition of independence has been violated, since the probability of selecting a student who takes Spanish changes if one or more previously-sampled students take Spanish. For example, if all 50 students in the sample are taking Spanish, then the probability of picking someone who is enrolled in Spanish changes from for the first person to for the last one. Nevertheless, the binomial distribution is reasonably accurate because the sample is less than 10% of the population.PTS:121.ANS:A. Binomial distribution with n = 20 and p = 0.5. B. ; C. PTS:1 ................
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