Chapter 7 Review
AP Statistics
1. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5 you win
$1, if number of spots showing is 6 you win $4, and if the number of spots showing is 1, 2, or 3 you
win nothing. Let 𝑋 be the amount that you win when playing the game once. The expected value
of 𝑋 is
a. $0.00
b. $1.00
c. $2.50
d. $6.00
2. Which of the following is considered a discrete random variable?
a. The number of ounces of coke in a 12 oz. can.
b. The number of headaches experienced in a day at MVHS.
c. The amount of time it takes for a car to start.
d. The amount of carbon monoxide in the air (in moles).
e. All of the above are discrete.
3. 𝑋 and 𝑌 are independent random variables, and 𝑎 and 𝑏 are constants. Which one of the following
statements is true?
a. 𝜎𝑋+𝑌 = 𝜎𝑋 + 𝜎𝑌
2
b. 𝜎𝑋−𝑌
= 𝜎𝑋2 + 𝜎𝑌2
2
c. 𝜎𝑎+𝑏𝑋
= 𝑏𝜎𝑋2
d. 𝜎𝑋−𝑌 = 𝜎𝑋 + 𝜎𝑌
2
e. 𝜎𝑋+𝑌
= √𝜎𝑋2 + 𝜎𝑌2
4. A business evaluates a proposed venture as follows. It stand to make a profit of $10, 000 with
probability3⁄20, to make a profit of $5000 with probability 9⁄20, to break even with probability
5⁄ , and to lose $5000 with probability 3⁄ . The expected profit in dollars is
20
20
a. 1500
b. 0
c. 3000
d. 3250
e. -1500
5. A psychologist studied the number of puzzles subjects were able to solve in a five-minute period
while listening to soothing music. Let 𝑋 be the number of puzzles completed successfully by a
subject. 𝑋 had the following distribution:
𝑋
𝑃(𝑋)
1
2
3
4
0.2
0.4
0.3
0.1
Using the above data, the mean of 𝑋 is
a. 2.0
b. 2.3
c. 2.5
d. 3.0
e. The answer cannot be computed.
6. Billie decides to participate in one of two games of chance involving dice. In the first game each
player gets a chance to throw the dice 100 times and wins if the dice show a sum of 7 between
15% and 20% of the time. In the second game, he gets to toss the dice 125 times and wins if his
dice show a sum of 6 or 8 between 30% and 35% of the time. Billie decides to play the first game
because he has a better chance of winning in that game, according to
a. the Law of Large Numbers
b. the Law of Averages
c. the Central Limit Theorem
d. Chebyshev’s Theorem
e. Bayes’ Theorem
7. The Law of Large Numbers says that as the number of observations drawn increase, the mean 𝑥̅ of
the observed values eventually approaches the
a. Mean 𝜇 of the population
b. Mean 𝑥̅ of the population
c. Median of the population
d. Variance of the population
e. Standard deviation of the population
8. Early studies of probability were conducted by the Italian mathematician Girolamo Cardano
(1501-1576). One of the many dice games that Cardano studied was played with six 6-sided dice.
Each of these six dice had five blank faces and one face with a number. The number 1 through 6
each appeared on one of the six dice. All six dice were rolled at once, and the payoff to the gambler
was based on the sum of the numbers showing on the up faces. What is the expected value of the
sum obtained by rolling all 6 dice?
a. 3
b. 3.5
c. 6
d. 36
e. It cannot be determined.
9. If 𝑋 is a random variable, which of the following does NOT represent a probability distribution of
𝑋?
𝑋
0
1
2
3
a.
𝑃(𝑋) .1
.35
.40
.15
b.
c.
d.
e.
𝑋
𝑃(𝑋)
𝑋
𝑃(𝑋)
𝑋
𝑃(𝑋)
𝑋
𝑃(𝑋)
-2
.13
1
-1
.28
2
1
6
0
.1
1
.28
3
1
6
1
.2
1
.1
0
.18
4
1
6
4
.3
3
.2
2
.13
5
6
1
6
1
6
7
.4
9
.5
9
.4
5
.3
1
6
10. A company that rents DVDs from vending machines in grocery stores has developed the following
probability distribution for the random variable 𝑋 = the number of DVDs a customer rents per
visit to a machine.
𝑋
1
2
3
4
𝑃(𝑋)
0.5
0.3
0.1
0.1
a. Find and interpret the mean (expected value) of 𝑋.
b. Find and interpret the standard deviation of 𝑋.
c. Suppose the profit the company makes on each customer visit is $0.75 per DVD minus $0.05
“fixed costs.” That is, if 𝑃 =profit, then 𝑃 = 0.75𝑋 − 0.05. Use a linear transformation (𝑎 +
𝑏𝑋) of your results in (a) and (b) to find the mean and standard deviation for 𝑃.
11. ACT scores for the 1,171,460 members of the 2004 high school graduating class who took the test
closely followed the Normal distribution with mean 20.9 and standard deviation 4.8.
a. Find the probability a student’s score is greater than 30.
b. Find the probability a student’s score is between 20 and 30.
Use the following data for 12 and 13.
X
P(X)
1
0.2
2
0.5
𝜇𝑋 = 2.7 𝜎𝑋 = 1.55
5
0.3
Y
P(Y)
2
0.7
4
0.3
𝜇𝑌 = 2.6, 𝜎𝑌 = 0.917
12. Let the random variable 𝑇 = 𝑋 + 𝑌.
a. Find all possible values of 𝑇. Compute the probability that 𝑇 takes each of these values.
Summarize the probability distribution of 𝑇 in a table.
b. Show that the mean of 𝑇 is equal to 𝜇𝑋 + 𝜇𝑌 .
c. Confirm that the variance of 𝑇 is equal to 𝜎𝑋2 + 𝜎𝑌2 . Show that 𝜎𝑇 ≠ 𝜎𝑋 + 𝜎𝑌 .
13. Let the random variable 𝐷 = 𝑋 − 𝑌.
a. Find all possible values of 𝐷. Compute the probability that 𝐷 takes each of these values.
Summarize the probability distribution of 𝐷 in a table.
b. Show that the mean of 𝐷 is equal to 𝜇𝑋 − 𝜇𝑌 .
c. Confirm that the variance of 𝐷 is equal to 𝜎𝑋2 + 𝜎𝑌2 .
14. You have two scales for measuring weights in a chemistry lab. Both scales give answers that vary
a bit in repeated weighing of the same item. If the true weight of a compound is 2.00 g, the first
scale produces reading 𝑋 that have mean 2.000 g and standard deviation 0.002 g. The seconds
scale’s readings 𝑌 have mean 2.001 g and standard deviation 0.001 g. The readings 𝑋 and 𝑌 are
independent. Find the mean and standard deviation of the difference 𝑌 − 𝑋 between the readings.
Interpret each value in context.
Also do the following problems from the book: 7.12 and 7.13 on page 477 7.60 on page 507