AP Statistics - RV: Random Variables

1 Probability > Random variables · Level 3

Which of the following are true statements? I. By the law of large numbers, the mean of a random variable will get closer and closer to a specific value. II. The standard deviation of a random variable is never negative. III. The standard deviation of a random variable is 0 only if the random variable takes a lone single value.

A

I and II only

B

I and III only

C

II and III only

D

I, II, and III

E

None of the above gives the complete set of true responses.

2 Probability > Random variables · Level 3

A carnival game has three prizes with the following probabilities:

Prize ($)	0	25	50
Probability	0.8	0.16	0.04

What is the variance of the prize variable?

A

\$300

B

\$450

C

\$7,500

D

\$11,250

E

\$18,750

3 Probability > Random variables · Level 3

Box A has 3 \$20 bills and a single \$100 bill. Box B has 100 \$10 bills and 300 \$50 bills. Box C has 40 \$1 bills. You can have all of Box C or blindly pick one bill out of either Box A or Box B. Which choice offers the greatest expected winning?

A

Box A

B

Box B

C

Box C

D

Either Box A or Box B, but not Box C

E

All boxes offer the same expected winning.

4 Probability > Random variables · Level 3

An Ivy League college has an acceptance rate of 8 percent. In a random sample of 50 applications, what is the expected number of applicants that will be turned down?

A

$50(0.08)$

B

$50(0.92)$

C

$50(0.08)(0.92)$

D

$\sqrt{50(0.08)(0.92)}$

E

$\dfrac{\sqrt{(0.08)(0.92)}}{50}$

5 Probability > Random variables · Level 3

The number of condos a real estate agent sells monthly has the following probability distribution:

Number of Condos	0	1	2	3
Probability	0.37	0.28	0.25	0.10

The agent averages a commission of \$6,500 per sale. What is the expected monthly commission from selling condos?

A

$4 \times 6500(0.37 + 0.28 + 0.25 + 0.10)$

B

$6500[0(0.37) + 1(0.28) + 2(0.25) + 3(0.10)]$

C

$4 \times 6500[0(0.37) + 1(0.28) + 2(0.25) + 3(0.10)]$

D

$\dfrac{6500(0.37+0.28+0.25+0.10)}{4}$

E

$\dfrac{6500[0(0.37) + 1(0.28) + 2(0.25) + 3(0.10)]}{4}$

6 Probability > Random variables · Level 3

A company has a choice of three investment schemes. Option I gives a sure \$50,000 return on investment. Option II gives a 50% chance of returning \$70,000 and a 50% chance of returning nothing. Option III gives a 10% chance of returning \$90,000 and a 90% chance of returning nothing. Which option should the company choose?

A

Option I if it wants to maximize expected return.

B

Option II if it needs at least \$60,000 to pay off an overdue loan.

C

Option III if it needs at least \$80,000 to pay off an overdue loan.

D

All of the above answers are correct.

E

Because of chance, it really doesn't matter which option the company chooses.

7 Probability > Random variables · Level 3

Two fair coins are tossed. If both land on heads, the player wins \$5. If exactly one lands on heads, the player wins \$2. If it costs \$3 to play, what is the player's expected outcome after six games?

A

Loss of \$0.75

B

Loss of \$1.50

C

Loss of \$4.50

D

Win of \$2.25

E

Win of \$13.50

8 Probability > Random variables · Level 3

The number of days a juror serves on a grand jury can be considered a random variable. The table below shows the relative frequency distribution for the number of days someone serves on a federal grand jury.

Number of Days	1	2	3	4	5
Relative Frequency	0.3	0.4	0.15	0.1	0.05

Federal jurors are paid \$40 per day. Based on the above distribution, what is the mean amount of pay federal jurors receive for their service?

A

\$2.20

B

\$17.60

C

\$40

D

\$88

E

\$200

9 Probability > Random variables · Level 3

A carnival game charges 25 cents per minute while a player is trying to win a prize. The player will receive prizes of different values. However, it is very difficult to win, and prizes are actually independent of the number of minutes played. The average number of minutes a player tries before giving up has a mean of 8 and a standard deviation of 2. The average value of the prize a player receives has a value of \$1.75 with a standard deviation of \$0.50. What are the expected value and standard deviation for playing this game for the player?

A

Expected value = –$0.25 and standard deviation = $0.56

B

Expected value = –$0.25 and standard deviation = $0.71

C

Expected value = $1.50 and standard deviation = $0.56

D

Expected value = $1.50 and standard deviation = $0.71

E

Expected value = $1.50 and standard deviation = $0.75

10 Probability > Random variables · Level 3

The four sides of a tetrahedron are labeled 1 through 4. For a certain carnival game, the player wins the number of points on the face-down side when the tetrahedron is tossed. Unbeknownst to the player, the tetrahedron is weighted so that the side with a 1 is twice as likely to land facedown as any of the other sides, which all have equal probabilities. What are the mean and variance of the number of points a player should expect from one toss of the polyhedron?

A

Mean is 2.5 and variance is 1.118

B

Mean is 2.5 and variance is 1.291

C

Mean is 2.2 and variance is 1.166

D

Mean is 2.2 and variance is 1.36

E

Mean is 2.2 and variance is 1.4

11 Probability > Random variables · Level 3

Given two independent random variables, X with mean 28.1 and standard deviation 3.4, and Y with mean 23.7 and standard deviation 2.9, which of the following is a true statement?

A

The mean of X – Y is 51.8.

B

The median of X – Y is 4.4.

C

The range of X – Y is 51.8.

D

The standard deviation of X – Y is 6.3.

E

The variance of X – Y is 19.97.

12 Probability > Random variables · Level 3

Suppose X and Y are random variables with $\mu_x = 45$, $\sigma_x = 12$, $\mu_y = 42$, and $\sigma_y = 5$. Given that X and Y are independent, what is the standard deviation of the random variable X – Y?

A

$\sqrt{7}$

B

$\sqrt{17}$

C

7

D

13

E

169

13 Probability > Random variables · Level 3

Given a random variable X taking three possible values, $x_1, x_2, x_3$, which of the following is a true statement?

A

$x_1 + x_2 + x_3 = 1$

B

$E(X) = \dfrac{1}{3}\sum x_i$

C

$\text{var}(X) = \dfrac{1}{3}\sum(x_i - \overline{x})^2$

D

$E(X + c) = E(X) + c$

E

$\text{var}(aX) = a \cdot \text{var}(X)$

14 Probability > Random variables · Level 3

A store sells metallic, printed, and glow-in-the-dark fidget spinners for $11.98, $1.47, and \$5.40, respectively. If metallic, printed, and glow-in-the-dark fidget spinners represent 10 percent, 60 percent, and 30 percent, respectively, of the fidget spinner sales, what is the expected monetary value of one day's sales if 200 fidgets are sold?

A

$11.98(0.10) + 1.47(0.60) + 5.40(0.30)$

B

$200[3(11.98 + 1.47 + 5.40)]$

C

$200[11.98(0.10) + 1.47(0.60) + 5.40(0.30)]$ $\dfrac{200(11.98+1.47+5.40)}{3}$

D

$11.98(0.10)+1.47(0.60)+5.40(0.30)$

E

$\dfrac{200}{200}$

15 Probability > Random variables · Level 3

Suppose $X$ and $Y$ are random variables with $E(X) = 29$, $\text{var}(X) = 7$, $E(Y) = 35$, and $\text{var}(Y) = 9$. What are the expected value and variance of the random variable $X + Y$?

A

$E(X + Y) = 32$, $\text{var}(X + Y) = 8$

B

$E(X + Y) = 64$, $\text{var}(X + Y) = 8$

C

$E(X + Y) = 64$, $\text{var}(X + Y) = 16$

D

$E(X + Y) = 64$, $\text{var}(X + Y) = \sqrt{7^2 + 9^2}$

E

There is insufficient information to answer this question.

16 Probability > Random variables · Level 3

Suppose $X$ and $Y$ are independent random variables with $E(X) = 34$, $\text{var}(X) = 4$, $E(Y) = 23$, and $\text{var}(Y) = 3$. What is the variance of the random variable $X - Y$?

A

$4 - 3$

B

$4 + 3$

C

$4^2 - 3^2$

D

$4^2 + 3^2$

E

$\sqrt{4^2 + 3^2}$

17 Probability > Random variables · Level 3

Suppose $X$ and $Y$ are independent random variables, both with normal distributions. If $X$ has a mean of 45 with a standard deviation of 4 and if $Y$ has a mean of 35 with a standard deviation of 3, what is the probability that a randomly generated value of $X$ is greater than a randomly generated value of $Y$?

A

$P(z > -2)$

B

$P(z > -1)$

C

$P(z > 0)$

D

$P(z > 1)$

E

$P(z > 2)$

18 Probability > Random variables · Level 3

The random variable $X$ has a mean of 15 and a standard deviation of 10. The random variable $Y$ is defined by $Y = 5 + 3X$. What are the mean and standard deviation of $Y$?

A

The mean is 45, and the standard deviation is 30.

B

The mean is 45, and the standard deviation is 35.

C

The mean is 50, and the standard deviation is 10.

D

The mean is 50, and the standard deviation is 30.

E

The mean is 50, and the standard deviation is 35.

19 Probability > Random variables · Level 3

A random variable $X$ has a mean of 25 and a standard deviation of 8. The random variable $X - Y$ has a mean of 9 and a standard deviation of 10. Assuming $X$ and $Y$ are independent, what are the mean and standard deviation of the random variable $Y$?

A

$\mu_Y = 16$ and $\sigma_Y = 2$

B

$\mu_Y = 16$ and $\sigma_Y = 6$

C

$\mu_Y = 16$ and $\sigma_Y = 12.8$

D

$\mu_Y = 34$ and $\sigma_Y = 2$

E

$\mu_Y = 34$ and $\sigma_Y = 6$

20 Probability > Random variables · Level 3

Bowler A bowls an average of 225 with a standard deviation of 15. Bowler B bowls an average of 190 with a standard deviation of 20. Together they form a team with an average total score of $225 + 190 = 415$. Under what conditions is the standard deviation of their total score equal to $\sqrt{15^2 + 20^2} = 25$?

A

No conditions are required. The standard deviation is 25.

B

No matter what conditions are given, the standard deviation is different from 25.

C

As long as they consistently bowl their averages, the standard deviation is 25.

D

The standard deviation is 25 only if their scores are independent.

E

The standard deviation is 25 only if their scores are mutually exclusive.

21 Probability > Random variables · Level 3

The four sides of a tetrahedron are labeled 1 through 4. Let $X$ be the random variable whose values are the numbers that end up face down when the tetrahedron is tossed. Assume this is a fair tetrahedron, that is, all sides are equally likely to end up face down. The mean of $X$ is 2.5, and its variance is $\dfrac{5}{4}$. Let $Y$ be the random variable whose values are the differences between the face-down values of the first and second rolls of the tetrahedron if it is rolled twice. What is the standard deviation of $Y$?

A

$\dfrac{\sqrt{5}}{4}$

B

$\dfrac{\sqrt{5}}{4} + \dfrac{\sqrt{5}}{4}$

C

$\sqrt{\dfrac{5}{4} + \dfrac{5}{4}}$

D

$\sqrt{\dfrac{5}{4} - \dfrac{5}{4}}$

E

$\dfrac{5}{4} + \dfrac{5}{4}$

22 Probability > Random variables · Level 3

The variance of a set of data is 5. If every value in the set is multiplied by 4 and then added to 100, what is the new variance?

A

5

B

20

C

80

D

120

E

180

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