Deck 8: Random Variables

Question 1

Use the following information for questions:
For each situation, decide if the random variable described is a discrete random variable or a continuous random variable.

-Random variable X = the number of letters in a word picked at random out of the dictionary.

A) Discrete random variable
B) Continuous random variable

Answer 1

Discrete random variable

Question 2

Use the following information for questions:
For each situation, decide if the random variable described is a discrete random variable or a continuous random variable.

-Random variable X = the number of letters in the last name of a student picked at random from a class on English composition.

A) Discrete random variable
B) Continuous random variable

Answer 2

Discrete random variable

Question 3

Use the following information for questions:
For each situation, decide if the random variable described is a discrete random variable or a continuous random variable.

-Random variable X = the time (in seconds) it takes one email to travel between a sender and receiver.

A) Discrete random variable
B) Continuous random variable

Answer 3

Continuous random variable

Question 4

Use the following information for questions:
For each situation, decide if the random variable described is a discrete random variable or a continuous random variable.

-Random variable X = the weight (in pounds) a dieter will lose after following a two week weight loss program.

A) Discrete random variable
B) Continuous random variable

Question 5

Which one of these variables is a continuous random variable?

A) The time it takes a randomly selected student to complete an exam.
B) The number of tattoos a randomly selected person has.
C) The number of women taller than 68 inches in a random sample of 5 women.
D) The number of correct guesses on a multiple choice test.

Question 6

Which one of these 'X' variables is a discrete random variable?

A) An experiment in chemistry is repeated many times and X is the time required for a reaction to occur in seconds.
B) A student is randomly selected and X is the number of correct answers on a two question multiple-choice quiz.
C) A UPS package is randomly selected and X is the weight in pounds of the package.
D) A student is randomly selected and X is the distance they must travel in feet to go from their dorm room door to the door of their first class on Monday morning.

Question 7

Use the following information for questions:
The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below.

-What is the probability of getting 4 heads?

A) 1/16
B) 2/16
C) 4/16
D) Cannot be determined

Question 8

Use the following information for questions:
The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below.

-What is the probability of getting at least one head?

A) 1/16
B) 4/16
C) 5/16
D) 15/16

Question 9

Use the following information for questions:
The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below.

-What is the probability of getting 1 or 2 heads?

A) 4/16
B) 6/16
C) 10/16
D) 14/16

Question 10

Use the following information for questions:
The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below.

-What is the value of the cumulative distribution function at 3, i.e. P(X ? 3)?

A) 6/16
B) 10/16
C) 11/16
D) 15/16

Question 11

Use the following information for questions:
Based on her past experience, a professor knows that the probability distribution for X = number of students who come to her office hours on Wednesday is given below.

-What is the probability that at least 2 students come to office hours on Wednesday?

A) 0.50
B) 0.70
C) 0.80
D) 0.90

Question 12

Use the following information for questions:
Based on her past experience, a professor knows that the probability distribution for X = number of students who come to her office hours on Wednesday is given below.

-What is the value of the cumulative probability distribution at 2, i.e. P(X $\le$ 2)?

A) 0.50
B) 0.70
C) 0.80
D) 0.90

Question 13

Use the following information for questions:
Based on her past experience, a professor knows that the probability distribution for X = number of students who come to her office hours on Wednesday is given below.

-What is the probability that at least 1 student comes to office hours on Wednesday?

A) 0.50
B) 0.70
C) 0.80
D) 0.90

Question 14

Use the following information for questions:
In an experiment, a person guesses which one of three different cards a researcher has randomly picked (and hidden from the person who guesses). This is repeated four times, replacing the cards each time. Let X = number of correct guesses in the four tries. The probability distribution for X, assuming the person is just guessing, is partially provided below.
-What is the value of the missing probability P(X = 4)?

Question 15

Use the following information for questions:
In an experiment, a person guesses which one of three different cards a researcher has randomly picked (and hidden from the person who guesses). This is repeated four times, replacing the cards each time. Let X = number of correct guesses in the four tries. The probability distribution for X, assuming the person is just guessing, is partially provided below.
-What is the probability that the person would get 3 or more correct guesses?

Question 16

Use the following information for questions:
In an experiment, a person guesses which one of three different cards a researcher has randomly picked (and hidden from the person who guesses). This is repeated four times, replacing the cards each time. Let X = number of correct guesses in the four tries. The probability distribution for X, assuming the person is just guessing, is partially provided below.

-What is the value of P(X $\le$ 2) = probability that number of correct guesses is less than or equal to 2?

Question 17

Use the following information for questions:
In an experiment, a person guesses which one of three different cards a researcher has randomly picked (and hidden from the person who guesses). This is repeated four times, replacing the cards each time. Let X = number of correct guesses in the four tries. The probability distribution for X, assuming the person is just guessing, is partially provided below.
-Give the cumulative distribution function for the number of correct guesses.

Question 18

Use the following information for questions:
Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below.
-What is the probability that Ellen will get at least 2 A's?

Question 19

Use the following information for questions:
Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below.
-What is the value of the cumulative probability distribution at 3, i.e. P(X ≤ 3)?

Question 20

Use the following information for questions:
Joan has noticed that the probability distribution for X = number of students in line to use the campus ATM machine when she shows up to use it is shown below.

-What is the probability that there will be no more than 1 student in line when Joan shows up?

A) 0.10
B) 0.20
C) 0.70
D) 0.90

Question 21

Use the following information for questions:
Joan has noticed that the probability distribution for X = number of students in line to use the campus ATM machine when she shows up to use it is shown below.

-What is the expected value of X, E(X)?

A) 2.0
B) 2.2
C) 2.5
D) 3.0

Question 22

Use the following information for questions:
Joan has noticed that the probability distribution for X = number of students in line to use the campus ATM machine when she shows up to use it is shown below.

-The variance of X, V(X) = 1.16. What is the standard deviation of X?

A) 1.08
B) 1.16
C) 1.35
D) 2.20

Question 23

The payoff (X) for a lottery game has the following probability distribution. What is the expected payoff?

A) $0
B) $0.50
C) $1.00
D) $2.50

Question 24

The expected value of a random variable is the

A) value that has the highest probability of occurring.
B) mean value over an infinite number of observations of the variable.
C) largest value that will ever occur.
D) most common value over an infinite number of observations of the variable.

Question 25

In a gambling game, on every play, there is a 0.1 probability that you win $7 and a 0.9 probability that you lose $1. What is the expected value of this game?

A) +$2
B) $0.20
C) -$2
D) -$0.20

Question 26

Suppose that for X = net amount won or lost in a lottery game, the expected value is E(X) = -$0.50. What is the correct interpretation of this value?

A) The most likely outcome of a single play is a net loss of 50 cents.
B) A player will have a net loss of 50 cents every single time he or she plays this lottery game.
C) Over a large number of plays the average outcome for plays is a net loss of 50 cents.
D) A mistake must have been made because it's impossible for an expected value to be negative

Question 27

The formula for the standard deviation for any discrete random variables with values x_i and corresponding probabilities p_i is:

A)
B)
C)

Question 28

What characteristic of a random variable is described by the expected value?

A) Standard deviation
B) Mean
C) Most likely value
D) Maximum value

Question 29

The following probability distribution is for the random variable X = number of classes for which full time students at a university are enrolled in a semester: What is the mean number (expected value) of courses taken per student?

A) 4
B) 5
C) 5.2
D) 5.5

Question 30

The expected value for a random variable is

A) the long-run average.
B) the most likely value.
C) the most frequent value observed in a random sample of observations of the random variable.
D) always np.

Question 31

The mean for a population of N values is equivalent to

A) the mean of any random sample taken from the population.
B) the median of the population of N values.
C) the expected value of a random variable defined as X = value for a randomly sampled individual from the population.
D) Np where p = proportion of values in the population that are unique.

Question 32

Use the following information for questions:
Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below.

-What is the expected number of A's she will get? (i.e. What is E(X)?)

Question 33

Use the following information for questions:
Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below.
-What is the variance for the number of A's she will get? (i.e. What is V(X)?)

Question 34

Use the following information for questions:
Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below.

-What is the standard deviation for the number of A's she will get?

Question 35

Use the following information for questions:
The Southside Bowling Alley has collected data on the number of children that come to birthday parties held at the bowling alley. Let the random variable X = the number of children per party. The distribution for the random variable X is given below.
-What is the probability that at least 7 children will come to a party?

Question 36

Use the following information for questions:
The Southside Bowling Alley has collected data on the number of children that come to birthday parties held at the bowling alley. Let the random variable X = the number of children per party. The distribution for the random variable X is given below.
-Suppose one party is to be randomly selected. We know that there will be at least 7 children at this party. What is the probability that there will be 10 children at the party?

Question 37

Use the following information for questions:
The Southside Bowling Alley has collected data on the number of children that come to birthday parties held at the bowling alley. Let the random variable X = the number of children per party. The distribution for the random variable X is given below.

-Suppose one party is to be randomly selected. What is the expected number of children that will attend this party? Include the appropriate symbol and units in your answer.

Question 38

Use the following information for questions:
The Southside Bowling Alley has collected data on the number of children that come to birthday parties held at the bowling alley. Let the random variable X = the number of children per party. The distribution for the random variable X is given below.

-What is the standard deviation for the number of children per party? Include the appropriate symbol and units in your answer.

Question 39

Use the following information for questions:
Did high gas prices keep Americans from hitting the road this past summer? In a nationwide survey of adults, one variable measured was how many days vacationers spent driving on the road on their longest trip. Consider the following (partial) probability distribution for the random variable X = the number of days for the longest car trip.
-Suppose the probability of 7 days is twice as likely as the probability of 8 days. What are the two missing probabilities to complete the distribution for X?

Question 40

Use the following information for questions:
Did high gas prices keep Americans from hitting the road this past summer? In a nationwide survey of adults, one variable measured was how many days vacationers spent driving on the road on their longest trip. Consider the following (partial) probability distribution for the random variable X = the number of days for the longest car trip.
-What is the expected number of days for the longest trip? Include symbol, value, and units.

Question 41

Which one of these variables is a binomial random variable?

A) Time it takes a randomly selected student to complete a multiple choice exam.
B) Number of textbooks a randomly selected student bought this term.
C) Number of women taller than 68 inches in a random sample of 5 women.
D) Number of CDs a randomly selected person owns.

Question 42

Consider an experiment that involves repeatedly rolling a six-sided die. Which of the following is a binomial random variable?

A) The number of rolls until a "4" is rolled for the first time.
B) The number of times that a "4" is rolled when the die is rolled six times.
C) The sum of the numbers observed on the first six rolls.
D) It is not possible to have a binomial random variable when rolling a six-sided die because a binomial random variable allows only two possible outcomes, not six.

Question 43

Which of the following is an example of a binomial random variable?

A) The number of games your favorite baseball team will win this coming season.
B) The number of questions you would get correct on a multiple-choice test if you randomly guessed on all questions.
C) The number of siblings a randomly selected student has.
D) The number of coins a randomly selected student is carrying.

Question 44

A medication produces side effects in each user with probability 0.10 and this is independent from one person to the next. If 50 people use the medication, the number who will experience side effects is

A) a binomial random variable.
B) always 5.
C) always 10%.
D) the value for which the probability distribution function (pdf) has the largest value.

Question 45

The probability is p = 0.80 that a patient with a certain disease will be successfully treated with a new medical treatment. Suppose that the treatment is used on 40 patients. What is the "expected value"of the number of patients who are successfully treated?

A) 40
B) 20
C) 8
D) 32

Question 46

Sara is a frequent business traveler. For security purposes, 10% of all people boarding airplanes are randomly selected for additional screening just prior to boarding. Define the random variable X = number of flights Sara completes before being chosen for additional screening. For instance, if she is searched boarding her next flight, then X = 0. What is the value of P(X = 2) = probability Sara completes two flights without screening and then is chosen for additional screening on the next one?

A) (0.1)²
B) (0.9)²
C) (0.1)²(0.9)
D) (0.9)²(0.1)

Question 47

A landscaping company claims that 90% of the trees they plant survive (defined as being still alive one year from planting). If a tree does not survive, the company will replace the tree with a new one. A homeowner will have 5 trees planted in his yard by this landscaping company. Consider these 5 trees to be a random sample of all trees planted by this company. If the company's claim is correct, what is the probability that all 5 of the trees will survive?

A) (0.1)⁵
B) (0.9)⁵
C) (0.1) + (0.1) + (0.1) + (0.1) + (0.1) = 0.5
D) (0.9)

Question 48

Use the following information for questions:
Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next.

-What is the probability that she will find exactly 2 used books?

A) 0.060
B) 0.185
C) 0.324
D) 0.600

Question 49

Use the following information for questions:
Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next.

-What is the probability that she will find exactly 3 used books?

A) 0.060
B) 0.185
C) 0.324
D) 0.900

Question 50

Use the following information for questions:
Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next.

-What is the expected number of used books she will find, E(X)?

A) 1.8
B) 2.0
C) 3.0
D) 6..0

Question 51

Use the following information for questions:
Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next.

-What is the variance for the number of used books she will find, V(X)?

A) 0.54
B) 1.12
C) 1.26
D) 61.80

Question 52

Use the following information for questions:
A child is observing squirrels in the park and notices that some are black and some are gray. For the next five squirrels she sees, she counts X = the number of black squirrels. Suppose X is a binomial random variable with n = 5 and p = 0.50.

-What is the probability that she will see exactly one black squirrel out of the five?

A) 0.031
B) 0.156
C) 0.313
D) 0.500

Question 53

Use the following information for questions:
A child is observing squirrels in the park and notices that some are black and some are gray. For the next five squirrels she sees, she counts X = the number of black squirrels. Suppose X is a binomial random variable with n = 5 and p = 0.50.

-What is the probability that she will see exactly two black squirrels out of the five?

A) 0.031
B) 0.156
C) 0.313
D) 0.500

Question 54

Use the following information for questions:
A child is observing squirrels in the park and notices that some are black and some are gray. For the next five squirrels she sees, she counts X = the number of black squirrels. Suppose X is a binomial random variable with n = 5 and p = 0.50.

-What is the expected number of black squirrels she will see, E(X)?

A) 2.0
B) 2.5
C) 3.0
D) 3.5

Question 55

Use the following information for questions:
A child is observing squirrels in the park and notices that some are black and some are gray. For the next five squirrels she sees, she counts X = the number of black squirrels. Suppose X is a binomial random variable with n = 5 and p = 0.50.

-What is the variance for the number of black squirrels she will see, V(X)?

A) 1.00
B) 1.12
C) 1.25
D) 2.50

Question 56

Use the following information for questions:
In a family with 4 children, the number of children with blue eyes is a binomial random variable X with n = 4 and p = .20.
-What is the probability that all 4 children will have blue eyes?

Question 57

Use the following information for questions:
In a family with 4 children, the number of children with blue eyes is a binomial random variable X with n = 4 and p = .20.
-What is the probability that exactly 3 children will have blue eyes?

Question 58

Use the following information for questions:
In a family with 4 children, the number of children with blue eyes is a binomial random variable X with n = 4 and p = .20.
-What is the probability that none of the children will have blue eyes?

Question 59

Use the following information for questions:
In a family with 4 children, the number of children with blue eyes is a binomial random variable X with n = 4 and p = .20.
-What is the expected number of children with blue eyes, E(X)?

Question 60

Use the following information for questions:
In a family with 4 children, the number of children with blue eyes is a binomial random variable X with n = 4 and p = .20.
-What is the variance for the number of children with blue eyes, V(X)?

Question 61

A random variable cannot be both continuous and

A) discrete.
B) uniform.
C) normal.
D) skewed.

Question 62

Which one of the following probabilities is a cumulative probability?

A) The probability that there are exactly 4 people with Type O+ blood in a sample of 10 people.
B) The probability of exactly 3 heads in 6 flips of a coin.
C) The probability that the accumulated annual rainfall in a certain city next year, rounded to the nearest inch, will be 18 inches.
D) The probability that a randomly selected woman's height is 67 inches or less.

Question 63

Keeping in mind that a normal distribution is a model for a continuous variable, which one of the following variables cannot possibly have a normal distribution?

A) People's opinions about a new tax law (favor or oppose)
B) Weights of 5-year-old male children
C) Handspans of adult females
D) Ounces of soda in cans labeled as having 12 ounces

Question 64

The time taken to answer an exam question for a randomly chosen student has a uniform probability distribution from 1 minute to 5 minutes. What is the probability that the time to answer is no more than 2 minutes?

A) 0.20
B) 0.25
C) 0.40
D) 0.75

Question 65

The time taken to deliver a pizza has a uniform probability distribution from 20 minutes to 60 minutes. What is the probability that the time to deliver a pizza is at least 25 minutes?

A) 0.125
B) 0.300
C) 0.700
D) 0.875

Question 66

Use the following information for questions:
The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.

-What is the standardized score (z-score) for a boot-up time of x =30 seconds?

A) -2.0
B) 0.0
C) 1.0
D) 2.0

Question 67

Use the following information for questions:
The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.

-What is the standardized score (z-score) for a boot-up time of x =20 seconds?

A) -2.0
B) 0.0
C) 1.0
D) 2.0

Question 68

Use the following information for questions:
The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds.

-What is the standardized score (z-score) for a boot-up of time x =35 seconds?

A) -2.0
B) 0.0
C) 1.0
D) 2.0

Question 69

Use the following information for questions:
Find the requested probability for the standard normal random variable Z.

-What is the probability that Z is less than or equal to 2, P(Z $\leq$ 2)?

A) 0.0228
B) 0.2000
C) 0.5000
D) 0.9772

Question 70

Use the following information for questions:
Find the requested probability for the standard normal random variable Z.

-What is the probability that Z is greater than 2, P(Z > 2)?

A) 0.0228
B) 0.2000
C) 0.5000
D) 0.9772

Question 71

Use the following information for questions:
Find the requested probability for the standard normal random variable Z.

-What is the probability that Z is between -1 and 1, P(-1 $\leq$ Z $\leq$ 1)?

A) 0.1587
B) 0.3174
C) 0.6826
D) 0.8413

Question 72

Use the following information for questions:
Find the requested probability for the standard normal random variable Z.

-What is the probability that Z is between -1.2 and 1.45, P(-1.2 $\leq$ Z $\leq$ 1.45)?

A) 0.0303
B) 0.7740
C) 0.8041
D) 0.8114

Question 73

Assuming a standard normal distribution is appropriate, what is the approximate probability that a z-score is greater than or equal to 2.33? Said another way, what is P(Z $\geq$ 2.33)?

A) 0.99
B) 0.01
C) 0.15
D) 0.25

Question 74

Suppose that vehicle speeds at an interstate location have a normal distribution with a mean equal to 70 mph and standard deviation equal to 8 mph. What is the z-score for a speed of 64 mph?

A) -6
B) -0.75
C) +0.75
D) +6

Question 75

Heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. About what proportion of college women are between 65 and 67 inches tall?

A) 0.75
B) 0.50
C) 0.25
D) 0.17

Question 76

Verbal SAT scores have approximately a normal distribution with mean equal to 500 and standard deviation equal to 100. The 95^th percentile of z-scores is z = 1.65. What is the 95^th percentile of verbal SAT scores?

A) 335
B) 500
C) 600
D) 665

Question 77

Verbal SAT scores have a mean of 500 and a standard deviation of 100. Which of the following describes how to find the proportion of Verbal SAT scores that are greater than 600?

A) Find the area to the left of z = 1 under a standard normal curve.
B) Find the area between z =-1 and z = 1 under a standard normal curve.
C) Find the area to the right of z = 1 under a standard normal curve.
D) Find the area to the right of z = -1 under a standard normal curve.

Question 78

Pulse rates of adult men are approximately normal with a mean of 70 and a standard deviation of 8. Which choice correctly describes how to find the proportion of men that have a pulse rate greater than 78?

A) Find the area to the left of z = 1 under a standard normal curve.
B) Find the area between z = -1 and z = 1 under a standard normal curve.
C) Find the area to the right of z =1 under a standard normal curve.
D) Find the area to the right of z = - under a standard normal curve.

Question 79

Scores on an achievement test had an average of 70 and a standard deviation of 10. Serena's score was 85. Assuming the scores have approximately a normal distribution, about what proportion of students scored lower than Serena?

A) 0.93
B) 0.07
C) 0.84
D) 0.68

Question 80

Weights of females have approximately a normal distribution with mean 135 lbs. and standard deviation 20 lbs. Allison weighs 145 lbs. What is the z-score for her weight?

A) 10
B) 1.50
C) 0.50
D) 0.20