ap stats review
The random variable K has a geometric distribution with mean 16. Which of the following is closest to the standard deviation of random variable K ?
A- 0.0625 B- 0.9375 C- 4 D- 15.49 E- 240 Answer is D Since the formula for mean =1/p, rearranging the formula gives p=1/μ=1/16=0.0625p. Thus, the standard deviation of random variable KK is σK= (Sqrt (1−0.0625))/.0625 ≈15.49.
Approximately 9 percent of the residents of a large city have seen a certain theater production that is currently playing in the city. A marketing researcher will randomly select residents until one is found who has seen the production. What is the expected number of residents the researcher will need to ask to find someone who has seen the production?
A- 0.09 B- 0.30 C- 10.60 D- 11.00 E- 11.11 Answer is E Let random variable X represent the number of residents the researcher will need to ask to find someone who has seen the production. Random variable X has a geometric distribution. The expected value of random variable X is E[X]=1/p=1/.09≈11.11 people.
A high school theater club has 40 students, of whom 6 are left-handed. Two students from the club will be selected at random, one at a time without replacement. What is the probability that the 2 students selected will both be left-handed?
A-30/1600 B-30/1560 C-36/1600 D-6/40 E-1156/1600 Answer is B If FF represents the event that the first student selected is left-handed and SS represents the event that the second student selected is left-handed, then P(F∩S)=[P(F)][P(S|F)]=(640)(539)=301,560
Past records indicate that 15 percent of the flights for a certain airline are delayed. Suppose flights are randomly selected one at a time from all flights. Assume each selection is independent of another. Which of the following is closest to the probability that it will take 5 selections to find one flight that is delayed?
A- 0.0783 B- 0.0921 C- 0.4780 D- 0.5220 E- 0.5563 Answer is B Let random variable X represent the number chosen to find a delayed flight. The variable can be modeled with a geometric distribution. Thus, P(X=5)= 0.15(0.85^4)≈0.0783
In the United States, 75 percent of adults wear glasses or contact lenses. A random sample of 10 adults in the United States will be selected. Which of the following is closest to the probability that fewer than 8 of the selected adults wear glasses or contact lenses?
A- 0.10 B- 0.28 C- 0.47 D- 0.53 E- 0.76 Answer is C Let the random variable X represent the number of adults out of 10 who wear glasses or contact lenses. The random variable has a binomial distribution. Define success as wearing glasses or contact lenses with probability 0.75, and define failure as not wearing them with probability 0.25. The probability that fewer than 8 adults wear glasses or contact lenses is 1−P(X≥8)=1−[P(X=8)+P(X=9)+P(X=10)]= 1−[(10 8)(0.75)^8(0.25)^2+(10 9)(0.75)^9(0.25)^1+(10 10)(0.75)^10(0.25)0]≈0.47
According to a 2015 Census Bureau survey, 75,511 of the 822,959 residents of Baltimore County, Maryland, were enrolled in college. Consider a sample of 800 residents of Baltimore County, Maryland in 2015 selected at random. Which of the following is closest to the expected value of the number in the sample enrolled in college?
A- 8.17 B- 28.3 C- 66.7 D- 73.4 E- 94.4 Answer is D The distribution can be modeled with a binomial variable. The expected value is the mean for a binomial distribution given by np=800(75,511/822,959)
In the United States, the generation of people born between 1946 and 1964 are known as baby boomers, and the generation of people born between 1981 and 1996 are known as millennials. Currently, 18 percent of the population are baby boomers and 27 percent of the population are millennials. A random sample of 500 people will be selected. Let the random variable B represent the number of baby boomers in the sample, and let the random variable M represent the number of millennials in the sample. By how much will the mean of M exceed the mean of B ?
A- 9 B- 45 C- 90 D- 135 E- 225 Answer is B The mean of B is np=500(0.18)=90. The mean of M is np=500(0.27)=135. The difference of the means is 135−90=45.
(b) Calculate and interpret the expected value of X. Show your work.
The expected value of X is the mean of X. E(X)=3(0.15)+4(0.40)+5(0.25)+6(0.15)+7(0.05)=4.55 If Miguel plays the hole many times, his average score will be about 4.55.
(b) Suppose the specialist wants to know the number of suspicious transactions that will need to be reviewed until reaching the first transaction that will be blocked. (i) Define the random variable of interest and state how the variable is distributed. (ii) Determine the expected value of the random variable and interpret the expected value in context.
(i) Let the random variable X represent the number of reviews until the first block is found. X follows a geometric distribution. (ii) The expected value of X is its mean. The mean of a geometric distribution is given by the formula 1/p, where p is the probability of success. In this case p=0.4p=0.4 and 1/.4= 2.5. The specialist can expect to review 2.5 transactions per day, on average, until finding the first transaction that will be blocked.
(c) Consider a batch of 10 randomly selected suspicious transactions. Suppose the specialist wants to know the probability that 2 of the transactions will be blocked. (i) Define the random variable of interest and state how the variable is distributed. (ii) Find the probability that 2 transactions in the batch will be blocked. Show your work.
(i) Let the random variable Y represent the number of blocked transactions in a batch of 10 suspicious transactions. Y follows a binomial distribution. (ii) The probability that Y will equal 2 is given by P(Y=2)=(10 2)(0.4)^2(0.6)^8≈0.1209.
According to a 2018 survey, 74 percent of employed young adults expect to bring work on a vacation trip. A random sample of 20 employed young adults will be selected. What is the probability that 8 of the selected young adults will expect to bring work on a vacation trip?
A- (20 8)(0.26)^8(0.74)^12 B- (20 8)(0.74)^8(0.26)^12 C- (12 8)(0.26)^8(0.74)^12 D- (12 8)(0.74)^8(0.26)^12 E- (28 8)(0.26)^8 (0.74)^12 Answer is B Let the random variable XX represent the number of young adults in a sample of 20 who expect to bring work on a vacation trip. Random variable XX has a binomial distribution with success defined as selecting one who expects to take work, probability 0.74, and failure as selecting one who does not, probability 0.26. The probability of selecting 8 young adults who expect to take work on vacation is P(X=8)P(X=8) =(20 8)(0.74)^8(0.26)^12.
At Mike's favorite coffee shop, the coffee of the day is either a dark roast, a medium roast, or a light roast. From past experience, Mike knows that the probability of the coffee being a light roast is 0.15 and the probability of the coffee being a dark roast is 0.25. What is the probability of the coffee of the day not being a light roast or a dark roast on the next day that Mike visits the coffee shop?
A- .15 B- .25 C- .40 D- .60 E- .80 Answer is D The event of the coffee not being a light roast or a dark roast is the complement of the event of the coffee being either a light roast or a dark roast, which is the same as the coffee being a medium roast (since there are only three options). The probability of the coffee being either a light roast or a dark roast is 0.25+0.15=0.40, so the probability of the complement is 1−0.40=0.601−0.40=0.60.
A representative from a company that manufactures items for left-handed people will attend a large convention. The representative hopes to find a left-handed person at the convention to try out the items. The representative will select an attendee at random until a left-handed person is found. Assume each selection is independent of another. If 10 percent of the convention attendees are left-handed, what is the probability that the representative must select 4 attendees to find one who is left-handed?
A- 0.1(0.9^3) B- 0.1(0.9^4) C- 0.12(0.9^2) D- 0.9(0.1^3) E- 0.9(0.1^4) Answer is A Let the discrete random variable XX represent the number of people chosen to find a left-handed person. The random variable has a geometric distribution and P(X=4)=0.1(0.9^3)
A consumer group is investigating the number of flights at a certain airline that are overbooked. They conducted a simulation to estimate the probability of overbooked flights in the next 5 flights. The results of 1,000 trials are shown in the following histogram. Based on the histogram, what is the probability that at least 4 of the next 5 flights at the airline will be overbooked? 1-.007 2-.181 3-.317 4-.332 5-.114
A- 0.114 B- 0.332 C- 0.446 D- 0.500 E- 0.886 Answer is C The sum of the relative frequencies of the bars for 4 and 5 is 0.446
At a large high school 40 percent of the students walk to school, 32 percent of the students have been late to school at least once, and 37.5 percent of the students who walk to school have been late to school at least once. One student from the school will be selected at random. What is the probability that the student selected will be one who both walks to school and has been late to school at least once?
A- 0.12 B- 0.15 C- 0.1875 D- 0.345 E- 0.72 Answer is B If WW represents the event that the selected student walks to school and LL represents the event that the selected student has been late to school at least once, then by the conditional probability formula P(L|W)=P(L∩W)/P(W). Solving for the numerator gives P(L∩W)=[P(W)][P(L|W)]=(0.40)(0.375)=0.15
According to a 2016 survey, 6 percent of workers arrive to work between 6:45 A.M. and 7:00 A.M. Suppose 300 workers will be selected at random from all workers in 2016. Let the random variable W represent the number of workers in the sample who arrive to work between 6:45 A.M. and 7:00 A.M. Assuming the arrival times of workers are independent, which of the following is closest to the standard deviation of W ?
A- 0.24 B- 4.11 C- 4.24 D- 16.79 E- 16.92 Answer is B Random variable W has a binomial distribution. The standard deviation for random variable W is sqrt np(1−p)= sqrt 300(0.06)(0.94)
Amy has 12 brown golf tees, 8 white golf tees, 10 red golf tees, 6 blue golf tees, and 12 green golf tees in her golf bag. If she selects one of the tees from the bag at random, what is the probability that she selects a tee that is not brown or blue?
A- 3/8 B- 5/8 C- 21/32 D- 3/4 E-7/8 Answer is B ere are 48 total tees in the bag, and 30 of the tees are a color other than brown or blue, so the probability is 30/48=5/8.
A random sample of n people selected from a large population will be asked whether they have read a novel in the past year. Let the random variable R represent the number of people from the sample who answer yes. The variance of random variable R is 6. Assume the responses are independent of each other. If the proportion of people from the population who read a novel in the past year is 0.40, which of the following is the best interpretation of random variable R ?
A- A binomial variable with 15 independent trials B- A binomial variable with 25 independent trials C- A variable that is not binomial with 25 independent trials D- A binomial variable with 40 independent trials E- A variable that is not binomial with 40 independent trials Answer is B The variable is binomial because the selected person will answer yes or no, there are nn fixed trials, the probability of success (responding yes) is constant, and responses are independent. If n=25n=25, the variance is np(1−p)=25(0.4)(0.6)=6np(1−p)=25(0.4)(0.6)=6, which is the correct variance.
At a certain restaurant, 35 percent of the customers order the daily special each day. Assume that each day the customers arrive randomly and order independently. Let the random variable X represent the number of orders placed until the first daily special is ordered. The distribution of X is geometric and has an expected value of approximately 2.86. Which of the following is the best interpretation of the expected value?
A- Each day, 3 customers will order the daily special. B- Over many days, the average number of customers ordering the daily special is approximately 2.86. C- Over many days, it takes about 2.86 orders, on average, to be placed until the first daily special is ordered. D- For a random sample of the days, the average number of orders of the daily special will be 2.86. E- Each day, the ratio of the number of orders of the daily special to the number of orders of other menu items is about 2.86 to 1. Answer is C The mean of a probability distribution is the expected value—that is, the long-run average over repeated trials. Over time, the average number of orders placed until the daily special is ordered is approximately 2.86.
In a certain region, 10 percent of the homes have solar panels. A city official is investigating energy consumption for homes within the region. Each week, the city official selects a random sample of homes from the region. Let random variable Y represent the number of homes selected at random from the region until a home that has solar panels is selected. The random variable Y has a geometric distribution with a mean of 10. Which of the following is the best interpretation of the mean?
A- Each week, the number of homes with solar panels increases by 10. B- For a randomly selected week, it will take 10 homes before a home with solar panels is selected. C- The average number of solar panels per home is equal to 10. D- Over many weeks, the average number of homes with solar panels is 10. E- Over many weeks, it takes 10 homes, on average, before a home with solar panels is selected. Answer is E The mean is the long-run average over repeated trials. Over time, the average number of homes that need to be selected before a home with solar panels is selected is 10.
A financial analyst reports that for people who work in the finance industry, the probability that a randomly selected person will have a tattoo is 0.20. Which of the following is the best interpretation of the probability 0.20 ?
A- For all workers in the United States, 20% will work in finance. B- For all finance workers, 20% will have a tattoo. C- For all people with tattoos, 20% will work in finance. D- For a specific group of 5 finance workers, 1 will have a tattoo. E- For a specific group of 5 people with a tattoo, 1 will work in finance. Answer is B In the long run, 0.20 is the relative frequency at which a finance worker will have a tattoo.
A certain spinner is divided into 6 sectors of equal size, and the spinner is equally likely to land in any sector. Four of the 6 sectors are shaded, and the remaining sectors are not shaded. Which of the following is the best interpretation of the probability that one spin of the spinner will land in a shaded sector?
A- For many spins, the long-run relative frequency with which the spinner will land in a shaded sector is 1/3. B- For many spins, the long-run relative frequency with which the spinner will land in a shaded sector is 1/2. C- For many spins, the long-run relative frequency with which the spinner will land in a shaded sector is 2/3. D- For 6 spins, the spinner will land in a shaded sector 4 times. E- For 6 spins, the spinner will land in a shaded sector 2 times. Answer is C The spinner has 6 possible outcomes. There are 4 outcomes where the spinner will land in a shaded sector. In the long run, the relative frequency with which the spinner lands in a shaded sector is 2/3
A local department store estimates that 10 percent of its customers return the merchandise they purchase. Let the random variable R represent the number of returns for a random sample of 40 customers. Assume that random variable R follows a binomial distribution. What is described by the value of (40 8)(0.1)^8(0.9)^32 ?
A- The mean of the random variable B- The variance of the random variable C- The standard deviation of the random variable D- The probability that 8 customers in the sample will return merchandise E- The probability of selecting a sample of 40 customers who will all return merchandise Answer is D The value represents the probability for a binomial random variable of seeing 8 successes in 40 trials, where the probability of success is 0.1.
At a local elementary school, 35 percent of all students have brown eyes, 45 percent have brown hair, and 60 percent have brown hair or brown eyes. A student will be selected at random from the school. Let E represent the event that the selected person has brown eyes, and let H represent the event that the selected person has brown hair. Are E and H mutually exclusive events?
A- Yes, because P(E∩H)=0P(E∩H)=0. B- Yes, because P(E∩H)=0.2P(E∩H)=0.2. C- Yes, because P(E∩H)=0.6P(E∩H)=0.6. D- No, because P(E∩H)=0.2P(E∩H)=0.2. E- No, because P(E∩H)=0.6 Answer is D If EE represents the event that the selected student has brown eyes and HH represents the event that the selected student has brown hair, then P(E∩H)=P(E)+P(H)−P(E∪H)=0.35+0.45−0.60=0.20. Because the probability P(E∩H)P(E∩H) is not equal to 0, the events are not mutually exclusive.
A survey of people on pizza preferences indicated that 55 percent preferred pepperoni only, 30 percent preferred mushroom only, and 15 percent preferred something other than pepperoni and mushroom. Suppose one person who was surveyed will be selected at random. Let P represent the event that the selected person preferred pepperoni, and let M represent the event that the selected person preferred mushroom. Are P and M mutually exclusive events for the people in this survey?
A- Yes, because the joint probability of PP and MM is greater than 0. B- Yes, because the joint probability of PP and MM is greater than 1. C- Yes, because the joint probability of PP and MM is equal to 0. D- No, because the joint probability of PP and MM is equal to 1. E- No, because the joint probability of PP and MM is equal to 0. Answers is C The percentages of the pizza preferences for the people in this survey adds to 1. Therefore, a person cannot select both pepperoni and mushroom as his or her favorite pizza topping, because P(P∩M)=0P(P∩M)=0, which indicates the events are mutually exclusive.
Each person in a group of twenty people at a hotel orders one meal chosen from oatmeal, eggs, or pancakes and one hot beverage chosen from coffee or tea. One person will be selected at random from the twenty people. What is the sample space for the meal and beverage for the person selected?
A- {(oatmeal, coffee), (oatmeal, tea), (eggs, coffee), (eggs, tea), (pancakes, coffee), (pancakes, tea)} B- {(oatmeal, pancakes), (oatmeal, eggs), (eggs, pancakes), (coffee, tea)} C- {(coffee, tea, oatmeal), (coffee, tea, eggs), (coffee, tea, pancakes)} D- {oatmeal, coffee, pancakes, eggs, tea} E- {(oatmeal, eggs, pancakes), (coffee, tea)} Answer is A
At a financial institution, a fraud detection system identifies suspicious transactions and sends them to a specialist for review. The specialist reviews the transaction, the customer profile, and past history. If there is sufficient evidence of fraud, the transaction is blocked. Based on past history, the specialist blocks 40 percent of the suspicious transactions. Assume a suspicious transaction is independent of other suspicious transactions. (a) Suppose the specialist will review 136 suspicious transactions in one day. What is the expected number of blocked transactions by the specialist? Show your work.
If the fraud specialist reviews 136 transactions in a day, the expected number of blocks by the specialist is np=136(0.4)=54.4 blocks
Miguel is a golfer, and he plays on the same course each week. The following table shows the probability distribution for his score on one particular hole, known as the Water Hole. Score| 3| 4| 5| 6| 7| probability| .15| .40| .25| .15| .05| Let the random variable XX represent Miguel's score on the Water Hole. In golf, lower scores are better. (a) Suppose one of Miguel's scores from the Water Hole is selected at random. What is the probability that Miguel's score on the Water Hole is at most 5 ? Show your work.
P(X≤5)=0.15+0.40+0.25=0.80.
During a severe storm, electrical transformers that function independently are expected to operate 85 percent of the time. Suppose 20 electrical transformers are randomly selected from the population. Let the random variable T represent the number of electrical transformers operating during a severe storm. Which of the following is the best interpretation of the random variable T ?
A- It is a binomial variable with mean 17 transformers and standard deviation 2.55−−−−√2.55 transformers. B- It is a binomial variable with mean 17 severe storms and standard deviation 2.55−−−−√2.55 severe storms. C- It is a binomial variable with mean 0.85 transformer and standard deviation 20 transformers. D- It is a variable that is not binomial with mean 17 transformers and standard deviation 2.55−−−−√2.55 transformers. E- It is a variable that is not binomial with mean 0.85 severe storm and standard deviation 20 severe storms. Answer is B The variable is binomial because the selected transformer is either operating or not, there are 20 fixed trials, the probability of success (operating) is constant, and the transformers function independently. The mean is 20(0.85)=17 and the standard deviation is sqrt 20(0.85)(0.15) = sqrt 2.55. The unit associated with the mean and standard deviation is transformers.
In a certain population of birds, about 40 percent of the birds have a wingspan greater than 10 inches. Biologists studying the birds will create a simulation with random numbers to estimate the probability of finding 1 bird in a sample of 6 birds with a wingspan greater than 10 inches. Which of the following assignments of the digits 0 to 9 will model the population?
A- Let the even digits represent birds with a wingspan greater than 10 inches and the odd digits represent birds with a wingspan less than or equal to 10 inches. B- Let the digits 0 and 1 represent birds with a wingspan greater than 10 inches and the remaining digits represent birds with a wingspan less than or equal to 10 inches. C- Let the digits from 0 to 2 represent birds with a wingspan greater than 10 inches and the remaining digits represent birds with a wingspan less than or equal to 10 inches. D- Let the digits from 0 to 2 represent birds with a wingspan greater than 10 inches and the remaining digits represent birds with a wingspan less than or equal to 10 inches. E- Let the digits from 0 to 4 represent birds with a wingspan greater than 10 inches and the remaining digits represent birds with a wingspan less than or equal to 10 inches. The answer is D The four digits 0, 1, 2, and 3 represent 40% of the 10 digits and would model 40% of the population.
A business journal reports that the probability that Internet users in the United States will use a mobile payment app is 0.60. The journal claims this indicates that out of 5 randomly selected Internet users, 3 will use the mobile payment app. Is the business journal interpreting the probability correctly?
A- No, because the Internet users are not independent of each other. B- No, because only 60% of all people use the Internet. C- No, because 0.60 represents probability in the long run for many Internet users. D- Yes, because Internet users are selected at random. E- Yes, because 3 out of 5 is equal to 60%. The answer is C
Consider rolling two number cubes, each of which has its faces numbered from 1 to 6. The cubes will be rolled and the sum of the numbers landing face up will be recorded. Let the event E represent the event of rolling a sum of 5. How many outcomes are in the collection for event E ?
A- One B- Two C- Four D- Five E- Six Answer is C There are four outcomes that sum to 5: 1 and 4, 4 and 1, 2 and 3, 3 and 2.
