Deck 4: Discrete Random Variables

Question 1

Classify the following random variable according to whether it is discrete or continuous. The number of phone calls to the attendance office of a high school on any given school day

A) discrete
B) continuous

Answer 1

A

Question 2

Explain why the following is or is not a valid probability distribution for the discrete random variable x.

Answer 2

This is not a valid probability distribution because the sum of the probabilities is less than 1.

Question 3

Explain why the following is or is not a valid probability distribution for the discrete random variable x.

Answer 3

This is not a valid probability distribution because the sum of the probabilities is greater than 1.

Question 4

The school newspaper surveyed 100 commuter students and asked two questions. First, students were asked how many courses they were currently enrolled in. Second, the commuter students were asked to estimate how long it took them to drive to campus. Considering these two variables, number of courses would best be considered a _________ variable and drive time would be considered a _________ variable.

A) discrete; continuous
B) discrete; discrete
C) continuous; continuous
D) continuous; discrete

Question 5

Classify the following random variable according to whether it is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch

A) discrete
B) continuous

Question 6

The Fresh Oven Bakery knows that the number of pies it can sell varies from day to day. The owner believes that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at a cost of $2.50 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $3 each, find the probability distribution for her daily profit.

Question 7

Classify the following random variable according to whether it is discrete or continuous. The number of pills in a container of vitamins

A) discrete
B) continuous

Question 8

Classify the following random variable according to whether it is discrete or continuous. The blood pressures of a group of students the day before the final exam

A) continuous
B) discrete

Question 9

Explain why the following is or is not a valid probability distribution for the discrete random variable x.

Question 10

A coin is flipped 6 times. The variable x represents the number of tails obtained. List the possible values of x. Is x discrete or continuous? Explain.

Question 11

Explain why the following is or is not a valid probability distribution for the discrete random variable x.

Question 12

50 students were randomly sampled and asked questions about their exercise habits. One of the questions they were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week. This variable would be characterized as which type of random variable?

A) discrete
B) continuous

Question 13

Classify the following random variable according to whether it is discrete or continuous. The number of goals scored in a soccer game

A) discrete
B) continuous

Question 14

A bottle contains 16 ounces of water. The variable x represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of x? Is x discrete or continuous? Explain.

Question 15

Consider the given discrete probability distribution. Construct a graph for p(x).

Question 16

Classify the following random variable according to whether it is discrete or continuous. The height of a player on a basketball team

A) continuous
B) discrete

Question 17

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the probability for the value of x = 5. $$\begin{array} { c | c c c c c } \mathrm { x } & 2 & 3 & 5 & 8 & 10 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & ? ? ? & 0.30 & 0.10\end{array}$$

A) 0.7
B) 0.1
C) 0.3
D) 0.2

Question 18

Management at a home improvement store randomly selected 95 customers and observed their shopping habits. They recorded the number of items each of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the managers of the home improvement store.

A) number of items - discrete; total time - continuous
B) number of items - continuous; total time - continuous
C) number of items - continuous; total time - discrete
D) number of items - discrete; total time - discrete

Question 19

Classify the following random variable according to whether it is discrete or continuous. The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska

A) continuous
B) discrete

Question 20

Classify the following random variable according to whether it is discrete or continuous. The speed of a car on a Los Angeles freeway during rush hour traffic

A) continuous
B) discrete

Question 21

A dice game involves rolling three dice and betting on one of the six numbers that are on the dice. The game costs $11 to play, and you win if the number you bet appears on any of the dice. The distribution for the outcomes of the game (including the profit) is shown below: 0 -$11 125/216 $$\begin{array} { c c c } \hline \text { Number of dice with your number } & \text { Profit } & \text { Probability } \\\hline 0 & - \$ 11 & 125 / 216 \\1 & \$ 11 & 75 / 216 \\2 & \$ 13 & 15 / 216 \\3 & \$ 33 & 1 / 216 \\\hline\end{array}$$ Find your expected profit from playing this game.

A) -$1.53
B) $0.50
C) $11.20
D) $5.96

Question 22

Consider the given discrete probability distribution. Find P(x = 1 or x = 2).

Question 23

Consider the given discrete probability distribution. Find P(x ? 4). $$\begin{array} { | c | c | c | c | c | c | c | } \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p ( x ) & .30 & .25 & .20 & .15 & .05 & .05 \\\hline\end{array}$$

A) .95
B) .90
C) .05
D) .10

Question 24

The expected value of a discrete random variable must be one of the values in which the random variable can result.

Question 25

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below: $$\begin{array} { l | c c c } \text { Price } & \$ 10.00 & \$ 12.50 & \$ 15.00 \\\hline \text { Probability } & 0.25 & 0.45 & 0.3\end{array}$$ Suppose the mean cost of the frogs is $12.63 per week. Interpret this value.

A) The average cost for all weekly frog purchases is $12.63.
B) Most of the weeks resulted in frog costs of $12.63.
C) The median cost for the distribution of frog costs is $12.63.
D) The frog cost that occurs more often than any other is $12.63.

Question 26

Consider the given discrete probability distribution. Find P(x < 2 or x > 3).

Question 27

Consider the given discrete probability distribution. Find P(x > 3). $$\begin{array} { | c | c | c | c | c | c | } \hline x & 1 & 2 & 3 & 4 & 5 \\\hline p ( x ) & .1 & .2 & .2 & .3 & .2 \\\hline\end{array}$$

A) .5
B) .7
C) .2
D) .3

Question 28

Consider the given discrete probability distribution. Construct a graph for p(x).

Question 29

A local bakery has determined a probability distribution for the number of cheesecakes it sells in a given day. The distribution is as follows: $$\begin{array} { l | c c c c r } \text { Number sold in a day } & 0 & 5 & 10 & 15 & 20 \\\hline \text { Prob (Number sold) } & 0.14 & 0.16 & 0.23 & 0.17 & 0.3\end{array}$$ Find the number of cheesecakes that this local bakery expects to sell in a day.

A) 11.65
B) 11.79
C) 20
D) 10

Question 30

An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $156 and the cost of flying first class is $355. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.]

Question 31

Consider the given discrete probability distribution. Find the probability that x equals 5. $$\begin{array} { l | c c c c } x & 3 & 5 & 7 & 9 \\\hline P ( x ) & 0.33 & ? & 0.27 & 0.31\end{array}$$

A) 0.09
B) 0.91
C) 0.45
D) 4.55

Question 32

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below: $$\begin{array} { l | c c c } \text { Price } & \$ 10.00 & \$ 12.50 & \$ 15.00 \\\hline \text { Probability } & 0.4 & 0.45 & 0.15\end{array}$$ How much should the lab budget for next year's frog orders assuming this distribution does not change? (Hint: Find the expected price and assume 52 weeks per year.)

A) $617.50
B) $11.88
C) $1188.00
D) $3,211,000.00

Question 33

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the standard deviation of the distribution.

Question 34

An automobile insurance company estimates the following loss probabilities for the next year on a $25,000 sports car: Assuming the company will sell only a $500 deductible policy for this model (i.e., the owner covers the first $500 damage), how much annual premium should the company charge in order to average $620 profit per policy sold? 2 Find Mean, Variance, Standard Deviation

Question 35

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the mean of the distribution.

Question 36

In a pizza takeout restaurant, the following probability distribution was obtained for the number of toppings ordered on a large pizza. Find the mean and standard deviation for the random variable. $$\begin{array} { c | c } x & \mathrm { P } ( x ) \\\hline 0 & .30 \\1 & .40 \\2 & .20 \\3 & .06 \\4 & .04\end{array}$$

A) mean: 1.14; standard deviation: 1.04
B) mean: 1.54; standard deviation: 1.30
C) mean: 1.30; standard deviation: 2.38
D) mean: 1.30; standard deviation: 1.54

Question 37

The random variable x represents the number of boys in a family with three children. Assuming that births of boys and girls are equally likely, find the mean and standard deviation for the random variable x.

A) mean: 1.50; standard deviation: .87
B) mean: 2.25; standard deviation: .87
C) mean: 1.50; standard deviation: .76
D) mean: 2.25; standard deviation: .76

Question 38

Calculate the mean for the discrete probability distribution shown here. $$\begin{array} { c | c c c c } X & 1 & 4 & 8 & 11 \\\hline P ( X ) & 0.28 & 0.24 & 0.17 & 0.31\end{array}$$

A) 6.01
B) 6
C) 24
D) 1.5025

Question 39

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the probability that the random variable x is a value greater than 5. $$\begin{array} { c | c c c c c } \mathrm { x } & 2 & 3 & 5 & 8 & 10 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & 0.30 & 0.30 & 0.10\end{array}$$

A) 0.40
B) 0.70
C) 0.30
D) 0.60

Question 40

Mamma Temte bakes six pies each day at a cost of $2 each. On 39% of the days she sells only two pies. On 38% of the days, she sells 4 pies, and on the remaining 23% of the days, she sells all six pies. If Mama Temte sells her pies for $4 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.]

A) $2.72
B) $14.72
C) -$8.00
D) -$8.32

Question 41

Compute $\left( \begin{array} { l } 4 \\4\end{array} \right)$

A) 1
B) 4
C) 6
D) 16

Question 42

Compute

Question 43

If x is a binomial random variable, compute p(x) for n = 5, x = 1, p = 0.4.

A) 0.2592
B) 0.2411
C) 0.2929
D) 0.2722

Question 44

If x is a binomial random variable, compute p(x) for n = 5, x = 1, q = 0.8.

A) 0.4096
B) 0.0064
C) 0.3850
D) 0.0068

Question 45

The binomial distribution can be used to model the number of rare events that occur over a given time period.

Question 46

Calculate the mean for the discrete probability distribution shown here.

Question 47

Find the mean and standard deviation of the probability distribution for the random variable x, which represents the number of cars per household in a small town.

Question 48

Compute $\frac { 7 ! } { 3 ! ( 7 - 3 ) ! }$

A) 35
B) 210
C) 840
D) 70

Question 49

For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the probability of success is .4?

A) the probability of 4 failures in 5 trials where the probability of success is .6
B) the probability of 1 success in 5 trials where the probability of failure is .6
C) the probability of 4 failures in 5 trials where the probability of success is .4
D) the probability of 4 failures in 5 trials where the probability of failure is .6 .

Question 50

For a binomial distribution, if the probability of success is .48 on the first trial, what is the probability of failure on the second trial?

Question 51

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 42% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled and asked if they believe their company president has low ethical standards and their years of experience at the company. Could the probability distribution for the number of years of experience be modelled by a binomial probability distribution?

A) No, a binomial distribution requires only two possible outcomes for each experimental unit sampled.
B) Yes, the sample is a random and independent sample.
C) Yes, the sample size is n = 20.
D) No, the employees would not be considered independent in the present sample.

Question 52

Which binomial probability is represented on the screen below?

Question 53

For a binomial distribution, if the probability of success is .63 on the first trial, what is the probability of success on the second trial?

Question 54

Compute $\left( \begin{array} { l } 9 \\4\end{array} \right)$

A) 126
B) 84
C) 3024
D) 15,120

Question 55

Consider the given discrete probability distribution.

Question 56

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Which of the following is necessary for this problem to be analyzed using the binomial random variable? I. There are two outcomes possible for each of the 20 voters sampled. II. The outcomes of the 20 voters must be considered independent of one another. III. The probability a voter will actually vote is 0.70, the probability they won't is 0.30. A) I only B) II only C) III only D) I, II, and III

Question 57

Compute $\left( \begin{array} { l } 5 \\4\end{array} \right)$

A) 5
B) 1
C) 10
D) 20

Question 58

Compute $\left( \begin{array} { l } 5 \\0\end{array} \right)$

A) 1
B) 5
C) 10
D) undefined

Question 59

A binomial random variable is defined to be the number of units sampled until x successes is observed.

Question 60

Which binomial probability is represented on the screen below?

A) The probability of 2 successes in 8 trials where the probability of success is .3.
B) The probability of 8 failures in 2 trials where the probability of failure is .3.
C) The probability of 2 successes in 8 trials where the probability of failure is .3.
D) The probability of 8 successes in 2 trials where the probability of success is .3.

Question 61

A new drug is designed to reduce a person's blood pressure. Thirteen randomly selected hypertensive patients receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is untreated is 0.5. Then what is the probability of observing 11 or more blood pressure drops in a random sample of 13 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal places.

Question 62

If x is a binomial random variable, calculate σ2 for n = 70 and p = 0.2.

A) 11.2
B) 14
C) 3.347
D) 2.8

Question 63

If x is a binomial random variable, calculate μ for n = 70 and p = 0.4.

A) 28
B) 35
C) 2.8
D) 16.8

Question 64

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential election.

A) 0.392
B) 0.228
C) 0.772
D) 0.608
E) 0.887

Question 65

According to a published study, 1 in every 4 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty-five men and asked each whether he has been involved in a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic accident? Round to the nearest whole number.

A) 19
B) 4
C) 6
D) 25

Question 66

A literature professor decides to give a 15-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than .10. What score should be set as the lowest passing grade?

A) 11
B) 9
C) 12
D) 10

Question 67

We believe that 82% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. If the true percentage is really 82%, find the probability of observing 38 or more students who consider statistics to be an exciting subject. Round to six decimal places.

A) 0.004161
B) 0.000435
C) 0.003726
D) 0.995839

Question 68

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 42% of all employees believe their company president possesses low ethical standards. Assume that responses were randomly and independently collected. A president of a local company that employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many of her company's employees believe that she possesses low ethical standards?

A) 420
B) 42
C) 580
D) 958

Question 69

We believe that 90% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 24 students from the population and observed fewer than five in our sample who consider statistics to be an exciting subject. Make an inference about the belief that 90% of the students consider statistics to be an exciting subject.

A) The 90% number is too high. The real percentage is lower than 90%.
B) The 90% number is too low. The real percentage is higher than 90%.
C) The 90% number is exactly right.
D) It is impossible to make any inferences about the 90% number based on this information.

Question 70

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo.

A) 0.722
B) 0.515
C) 0.207
D) 0.218

Question 71

If x is a binomial random variable, calculate σ for n = 100 and p = 0.3. Round to three decimal places when necessary.

A) 4.583
B) 30
C) 21
D) 5.477

Question 72

An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 14 of these cars are independently sampled, what is the probability that more than half need new gas tanks?

Question 73

About 40% of the general population donate time and energy to community projects. Suppose 15 people have been randomly selected from a community and each asked whether he or she donates time and energy to community projects. Let x be the number who donate time and energy to community projects. Use a binomial probability table to find the probability that more than five of the 15 donate time and energy to community projects.

Question 74

The probability that an individual is left-handed is 0.16. In a class of 80 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth when necessary.

A) mean: 12.8; standard deviation: 3.28
B) mean: 80; standard deviation: 3.28
C) mean: 12.8; standard deviation: 3.58
D) mean: 80; standard deviation: 3.58

Question 75

A local newspaper claims that 70% of the items advertised in its classifieds section are sold within 1 week of the first appearance of the ad. To check the validity of the claim, the newspaper randomly selected n = 25 advertisements from last year's classifieds and contacted the people who placed the ads. They found that 16 of the 25 items sold within a week. Based on the newspaper's claim, is it likely to observe who sold their item within a week? Use a binomial probability table.

Question 76

According to a recent study, 1 in every 6 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse.

A) 0.188687
B) 0.062896
C) 0.125791
D) 0.807120

Question 77

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 44% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards. Round to six decimal places.

A) 0.437608
B) 0.285201
C) 0.669843
D) 0.809834

Question 78

According to a recent study, 1 in every 9 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that at least 2 of the women sampled have been the victim of domestic abuse. Round to six decimal places.

A) 0.782924
B) 0.246677
C) 0.536248
D) 0.217076

Question 79

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election.

A) 0.780
B) 0.714
C) 0.845
D) 0.649

Question 80

A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 20 adults over 50, what is the mean and standard deviation of the number who wear glasses? Round to the nearest hundredth when necessary.

A) mean: 12.6; standard deviation: 2.16
B) mean: 12.6; standard deviation: 3.55
C) mean: 7.4; standard deviation: 2.16
D) mean: 7.4; standard deviation: 3.55