🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning

The Practice of Statistics for AP*

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 6

Random Variables

Educators


Problem 1

Suppose you toss a fair coin 4 times.
Let $X=$ the number of heads you get.
(a) Find the probability distribution of $X .$
(b) Make a histogram of the probability distribution.
Describe what you see.
(c) Find $P(X \leq 3)$ and interpret the result.

Xiaomeng Z.
Numerade Educator

Problem 2

Pair-a-dice Suppose you roll a pair of fair, six-sided
dice. Let $T=$ the sum of the spots showing on the
up-faces.
(a) Find the probability distribution of $T$
(b) Make a histogram of the probability distribution.
Describe what you see.
(c) Find $P(T \geq 5)$ and interpret the result.

Xiaomeng Z.
Numerade Educator

Problem 3

Spell-checking Spell-checking software catches
"nonword errors," which result in a string of letters
that is not a word, as when "the" is typed as "teh."
When undergraduates are asked to write a 250 -word
essay (without spell-checking), the number X of
nonword errors has the following distribution:
$$
\begin{array}{ccccc}{\text { Value of } X :} & {0} & {1} & {2} & {3} & {4} \\ {\text { Probability: }} & {0.1} & {0.2} & {0.3} & {0.3} & {0.1} \\ \hline\end{array}
$$
(a) Write the event "at least one nonword error" in
terms of $X$ . What is the probability of this event?
(b) Describe the event $X \leq 2$ in words. What is its
probability? What is the probability that $X<2 ?$

Xiaomeng Z.
Numerade Educator

Problem 4

In an experiment on the behavior of young children, each subject is placed in an area
with five toys. Past experiments have shown that the probability distribution of the number X of toys played with by a randomly selected subject is as follows:

(a) Write the event "plays with at most two toys" in
terms of $\mathrm{X}$ . What is the probability of this event?
(b) Describe the event $X>3$ in words. What is its
probability? What is the probability that $X \geq 3 ?$

Xiaomeng Z.
Numerade Educator

Problem 5

Faked numbers in tax returns, invoices, or expense account claims often display patterns that
aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford’s law.5 Call the first digit of a randomly chosen record X for short. Benford’s law gives this probability model for X (note that a first digit can’t be 0):

First digit X: 12345678 9
Probability: 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

(a) Show that this is a legitimate probability distribution.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) Describe the event $X \geq 6$ in words. What is $P(X \geq 6) ?$
(d) Express the event "first digit is at most 5$"$ in terms of $X$ . What is the probability of this event?

Xiaomeng Z.
Numerade Educator

Problem 6

Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many
times did you go to an exercise or fitness center or work out?” Call the response Y for short. Based on a large sample survey, here is a probability model for the answer you will get:

Days: 01234567
Probability: 0.68 0.05 0.07 0.08 0.05 0.04 0.01 0.02

(a) Show that this is a legitimate probability distribution.
(b) Make a histogram of the probability distribution. Describe what you sec.
(c) Describe the event $Y<7$ in words. What is $P(Y<7) ?$
(d) Express the event" "worked out at least once" in terms of Y. What is the probability of this event?

Xiaomeng Z.
Numerade Educator

Problem 7

Refer to Exercise 5. The first digit of a randomly chosen expense account claim follows
Benford's law. Consider the events $A=$ first digit is 7 or greater and $B=$ first digit is odd.

(a) What outcomes make up the event $A$ ? What is $P(A) ?$
(b) What outcomes make up the event $B$ ? What is $P(B) ?$
(c) What outcomes make up the event "A or $B^{\prime \prime} ?$ What is $P(A \text { or } B) ?$ Why is this probability not equal to $P(A)+P(B) ?$

Xiaomeng Z.
Numerade Educator

Problem 8

Refer to Exercise 6. Consider the events $A=$ works out at least once and $B=$ works out less than 5 times per week.

(a) What outcomes make up the event $A$ ? What is $P(A) ?$
(b) What outcomes make up the event $B$ ? What is $P(B) ?$
(c) What outcomes make up the event "A and B"? What is $P(A \text { and } B) ?$ Why is this probability not equal to $P(A) \cdot P(B) ?$

Xiaomeng Z.
Numerade Educator

Problem 9

Keno is a favorite game in casinos, and similar games are popular with the states that operate
lotteries. Balls numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls
are chosen at random. Players select numbers by marking a card. The simplest of the many wagers available is "Mark 1 Number." Your payoff is $\$ 3$ on a $\$ 1$ bet if the number you select is one of those chosen. Because 20 of 80 numbers are chosen, your probability of winning is $20 / 80,$ or 0.25 . Let $X=$ the amount you gain on a single play of the game. (a) Make a table that shows the probability distribution of $X .$ (b) Compute the expected value of $\mathrm{X}$ . Explain what
this result means for the player.

Xiaomeng Z.
Numerade Educator

Problem 10

Suppose a homeowner spends $\$ 300$ for a home insurance policy that will pay out
$\$ 200,000$ if the home is destroyed by fire. Let $Y=$ the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is 0.0002 .
(a) Make a table that shows the probability distribution of Y.
(b) Compute the expected value of Y. Explain what this result means for the insurance company.

Xiaomeng Z.
Numerade Educator

Problem 11

Refer to Exercise 3. Calculate the mean of the random variable X and interpret this
result in context.

Xiaomeng Z.
Numerade Educator

Problem 12

Refer to Exercise 4. Calculate the mean of the random variable X and interpret this
result in context.

Xiaomeng Z.
Numerade Educator

Problem 13

A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from 1 to 9. In that case, the first digit Y of a randomly selected expense amount would have the probability distribution shown in the histogram.

(a) Explain why the mean of the random variable $Y$ is located at the solid red line in the figure.
(b) The first digits of randomly selected expense amounts actually follow Benford's law (Exercise 5 )
What's the expected value of the first digit? Explain how this information could be used to detect a fake expense report.
(c) What's $P(Y>6) ?$ According to Benford's law, what proportion of first digits in the employee's
expense amounts should be greater than 6$?$ How could this information be used to detect a fake
expense report?

Xiaomeng Z.
Numerade Educator

Problem 14

A life insurance company sells a term insurance policy to a 21-year-old male that pays $\$ 100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$ each year as payment for the insurance. The amount Y that the company earns on this policy is $\$ 250$ per year, less the $\$ 100,000$ that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$ profit earned by the company:

Age at death: 21 22 23 24 25 26 or more
Profit: ?$99,750 ?$99,500
Probability: 0.00183 0.00186 0.00189 0.00191 0.00193

(a) Copy the table onto your paper. Fill in the missing values of $Y .$
(b) Find the missing probability. Show your work.
(c) Calculate the mean $\mu_{Y}$ . Interpret this value in context.

Xiaomeng Z.
Numerade Educator

Problem 15

Refer to Exercise 3. Calculate and interpret the standard deviation of the random variable X. Show your work.

Sheryl E.
Numerade Educator

Problem 16

Refer to Exercise 4. Calculate and interpret the standard deviation of the random
variable X. Show your work.

Xiaomeng Z.
Numerade Educator

Problem 17

Refer to Exercise 13. It might also be possible to detect an employee’s fake
expense records by looking at the variability in the first digits of those expense amounts.
(a) Calculate the standard deviation $\sigma_{\mathrm{Y}}$ . This gives us an idea of how much variation we'd expect in the employee's expense records if he assumed that first digits from 1 to 9 were equally likely.
(b) Now calculate the standard deviation of first digits that follow Benford's law (Exercise 5). Would
using standard deviations be a good way to detect fraud? Explain.

Xiaomeng Z.
Numerade Educator

Problem 18

(a) It would be quite risky for you to insure the life of a 21 -year-old friend under the terms of Exercise
14. There is a high probability that your friend would live and you would gain $\$ 1250$ in premiums. But if he were to die, youl would lose almost $\$ 100,000$ . Explain carefully why selling insurance
is not risky for an insurance company that insures many thousands of 21 -year-old men.
(b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is, the riskier the investment. We can measure the great risk of insuring a single person's life in Exercise 14 by computing the standard deviation of the income $Y$ that the insurer will receive. Find $\sigma_{Y}$ using the distribution and mean found in Exercise 14 .

Xiaomeng Z.
Numerade Educator

Problem 19

How do rented housing units differ from units occupied by their owners? Here are the distributions of the number of rooms for owner occupied units and renter-occupied units in San Jose, California:

Number of Rooms
1 2 3 4 5 6 7 8 9 10
Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035
Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003

Let $X=$ the number of rooms in a randomly selected owner-occupied unit and $Y=$ the number of rooms in a randomly chosen renter-occupied unit.
(a) Make histograms suitable for comparing the probability distributions of $X$ and $Y .$ Describe any differences that you observe.
(b) Find the mean number of rooms for both types of housing unit. Explain why this difference makes sense.
(c) Find the standard deviations of both $X$ and $Y$ Explain why this difference makes sense.

Xiaomeng Z.
Numerade Educator

Problem 20

In government data, a houschold consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States:

Number of Persons
1 2345 6 7
Household probability 0.25 0.32 0.17 0.15 0.07 0.03 0.01
Family probability 0 0.42 0.23 0.21 0.09 0.03 0.02

Let $X=$ the number of people in a randomly selected U.S. household and $Y=$ the number of people in a randomly chosen U.S. family.
(a) Make histograms suitable for comparing the probability distributions of $X$ and $Y .$ Describe any differences that you observe.
(b) Find the mean for each random variable. Explain why this difference makes sense.
(c) Find the standard deviations of both $X$ and $Y .$ Explain why this difference makes sense.

Xiaomeng Z.
Numerade Educator

Problem 21

Random numbers Let $X$ be a number between 0 and 1 produced by a random number generator.
Assuming that the random variable $X$ has a uniform distribution, find the following probabilities:
(a) $P(X > 0.49)$
(b) $P(X \geq 0.49)$
(c) $P(0.19 \leq X < 0.37 \text { or } 0.84 < X \leq 1.27)$

Xiaomeng Z.
Numerade Educator

Problem 22

Random numbers Let Y be a number between 0
and 1 produced by a random number generator.
Assuming that the random variable Y has a uniform
distribution, find the following probabilities:
(a) $P(Y \leq 0.4)$
(b) $P(Y < 0.4)$
(c) $P(0.1 < Y \leq 0.15 \text { or } 0.77 \leq Y < 0.88)$

Xiaomeng Z.
Numerade Educator

Problem 23

ITBS scores The Normal distribution with mean $\mu=6.8$ and standard deviation $\sigma=1.6$ is a good description of the lowa Test of Basic Skills (ITBS) vocabulary scores of seventh-grade students in Gary, Indiana. Call the score of a randomly chosen student $X$ for short. Find $P(X \geq 9)$ and interpret the result. Follow the four-step process.

Xiaomeng Z.
Numerade Educator

Problem 24

Running a mile A study of $12,000$ able-bodied male students at the University of Illinois found
that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute. $^{8}$ Choose a student at random from this group and call his time for the mile Y. Find $P(Y<6)$ and interpret the result. Follow the four-step process.

Xiaomeng Z.
Numerade Educator

Problem 25

A sample survey contacted an SRS of 663 registered voters in Oregon shortly after an
clection and asked respondents whether they had voted. Voter records show that 56$\%$ of registered
voters had actually voted. We will see later that in repeated random samples of size 663 , the proportion in the sample who voted (call this proportion With vary according to the Normal distribution
with mean $\mu=0.56$ and standard deviation $\sigma=0.019$

(a) If the respondents answer truthfully, what is $\mathrm{P}(0.52 \leq V \leq 0.60)$ ? This is the probability that the sample proportion $V$ estimates the population proportion 0.56 within $\pm 0.04$
(b) In fact, 72$\%$ of the respondents said they had voted $(V=0.72) .$ If respondents answer truthfully, what is $P(V \geq 0.72) ?$ This probability is so small that it is good evidence that some people who did not vote claimed that they did vote.

Xiaomeng Z.
Numerade Educator

Problem 26

How many close friends do you have? An opinion poll asks this question of an SRS of 1100
adults. Suppose that the number of close friends adults claim to have varies from person to person with mean $\mu=9$ and standard deviation $\sigma=2.5$
We will see later that in repeated random samples of size 1100 , the mean response $\overline{x}$ will vary according to the Normal distribution with mean 9 and standard deviation 0.075 . What is $P(8.9 \leq \overline{x} \leq 9.1)$ , the probability that the sample result $\overline{x}$ estimates the population truth $\mu=9$ to within $\pm 0.1 ?$

Willis J.
Numerade Educator

Problem 27

Exercises 27 and 28 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars:

Number of cars X: 012345
Probability: 0.09 0.36 0.35 0.13 0.05 0.02

A housing company builds houses with two-car garages. What percent of households have more cars than the garage can hold?
(a) 13$\%$
(b) 20$\%$
(c) 45$\%$
(d) 55$\%$
(e) 80$\%$

Xiaomeng Z.
Numerade Educator

Problem 28

Exercises 27 and 28 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars:

Number of cars X: 012345
Probability: 0.09 0.36 0.35 0.13 0.05 0.02

What’s the expected number of cars in a randomly selected American household?
(a) Between 0 and 5
(b) 1.00
(c) 1.75
(d) 1.84
(e) 2.00

Xiaomeng Z.
Numerade Educator

Problem 29

A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: Draw one
card at random from the deck. You win $\$ 10$ if the card drawn is an ace. Otherwise, you lose $\$ 1 .$ If you make this wager very many times, what will be the mean amount you win?

(a) About $-\$ 1$ , because you will lose most of the time.
(b) About $\$ 9,$ because you win $\$ 10$ but lose only $\$ 1$ .
(c) About $-\$ 0.15 ;$ that is, on average you lose about 15 cents.
(d) About $\$ 0.77 ;$ that is, on average you win about 77 cents.
(e) About $\$ 0,$ because the random draw gives you a fair bet.

Xiaomeng Z.
Numerade Educator

Problem 30

The deck of 52 cards contains 13 hearts. Here is another wager: Draw one card at random from the
deck. If the card drawn is a heart, you win $\$ 2$ . Otherwise, you lose $\$ 1$ . Compare this wager (call it Wager 2 ) with that of the previous exercise (call it Wager 1 ). Which one should you prefer?
(a) Wager 1 , because it has a higher expected value.
(b) Wager $2,$ because it has a higher expected value.
(c) Wager 1 , because it has a higher probability of winning.
(d) Wager 2 , because it has a higher probability of winning.
(e) Both wagers are equally favorable.

Xiaomeng Z.
Numerade Educator

Problem 31

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Better readers? (1.3) Did students have higher reading scores after participating in the chess program?Give appropriate statistical evidence to support your answer.

Xiaomeng Z.
Numerade Educator

Problem 32

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Chess and reading (4.3) If the study found a statistically significant improvement in reading scores,
could you conclude that playing chess causes an increase in reading skills? Justify your answer.

Bryan M.
Numerade Educator

Problem 33

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Predicting posttest scores (3.2) What is the equation of the linear regression model relating posttest
and pretest scores? Define any variables used.

Xiaomeng Z.
Numerade Educator

Problem 34

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

How well does it fit? $(3.2)$ Discuss what $s, r^{2},$ and the residual plot tell you about this linear regression model.

Xiaomeng Z.
Numerade Educator