AP Statistics
Review: Distributions
Name_____________________
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State whether the variable is discrete or continuous
a. The number of goals scored in a soccer game.
b. The time it takes to get to school
c.
The age of the oldest teacher.
2. The probability of a battery being defective is 35%. If a lot of batteries are checked until a
defective battery is found, what is the probability that they had to check
a. 4 batteries?
b. Less than 4 batteries?
3. In a recent survey, 80% of the community favored building a police substation in their neighborhood.
If 15 citizens are selected, what is the mean number favoring the substation? What is the standard
deviation?
4. A new battery's voltage may be acceptable (A) or unacceptable (U). Suppose than 80% of all
batteries have acceptable voltages. Four batteries are selected at random.
a.
Let x denote the number of acceptable voltages. Find the probability distribution of x.
b. Let y denote the number of batteries that must be tested until an acceptable voltage is found.
Find the probability distribution of y.
6.
A mail-order computer software business has six telephone lines. Let x denote the number of lines in
use at a specified time. The probability distribution is as follows.
x
p(x)
0
1
2
3
4
5
6
0.1 0.15 0.2 0.25 0.2 0.06 0.04
Write each of the following events in terms of x, and then calculate the probability of each of the
following:
a. At most three lines are in use.
b. Between two and five lines are in use.
c.
Fewer than four lines are in use.
d. Find P(x=3 | x < 4).
e.
Calculate the mean and standard deviation.
f.
What is the probability that the number of lines in use is farther than 2 standard deviations from
the mean?
7. Suppose that a professor never dismisses class on time and x represents the amount of time (minutes)
the professor keeps the class over. The density curve is shown in the figure below.
Density
8
a.
Time
What is the probability that at most 3 minutes elapse before being dismissed?
b. What is the probability that he keeps them between 2 and 4 minutes?
8. A local motel has 100 rooms. The occupancy rate for the winter months is 60%. Find the probability
that
a. At least 70 rooms will be rented.
b. Less than 65 rooms will be rented
c.
Exactly 56 rooms will be rented.
9. A service repair shop offers two plans for repair service. Based on the distribution of hours shown
below, what is the average cost for plan A? plan B? Which is the best plan? Find the standard
deviation of both plans.
Plan A: $200 plus $5 per hour
Plan B: $50 per hour.
Hours
p(h)
0
1
2
3
4
5
0.2 0.27 0.23 0.15 0.09 0.06
10. Soaring insurance rates have made it difficult for many people to afford car insurance. Suppose that
16% of all those driving in a certain city are uninsured. Consider a random sample of 200 drivers.
a.
What is the mean value of the number who are uninsured, and what is the standard deviation of
the number who are uninsured?
b. What is the approximate probability that between 25 and 40 (inclusive) drivers in the sample were
uninsured?
c.
If you learned that more than 50 among the 200 drivers were uninsured, would you doubt the 16%
figure? Explain.
11. Eddie is running the balloon darts game at the school fair. He has blown up hundreds of balloons with
notes about prize tickets inside them. Twelve percent of the notes say “you win 5 tickets,” twenty
percent say “You win 3 tickets,” and the rest say “Sorry, try again!” After each play, he replaces the
popped balloon with another one bearing the same note. Let T = the number of tickets won by a
randomly selected player of this game. Find the expectation of the game.
12. Find the P(x < 7).
20 m
13. Research suggests that about 24% of 12-year-olds in the United States can pick out the state of
Colorado on a map.
a. What is the probability that you must sample exactly 5 twelve-year-olds to find the first one
who can pick out Colorado on a map?
b. What is the probability that you must sample 5 or more twelve-year-olds to find the first one
who can pick out Colorado on a map?
14. Find the P(4 < x < 7)
10 m
15. An online poll reported that 20% of respondents subscribe to the “five-second rule.” That is, they
would eat a piece of food that fell onto the kitchen floor if it was picked up within five seconds. Let’s
assume this figure is accurate for the entire U. S. population, and we select 15 people at random from
this population.
a. Determine the probability that exactly 7 of the 15 people subscribe to the “five-second rule.”
(Use the formula)
b. Find the probability that at least 1 person out of the 15 will subscribe to the “five-second
rule.” (Use the formula)
c.
Find the probability that less than 3 people out of 15 subscribe to the “five-second rule.” (Use
the formula).
16. Find the following given the data to the right.
a.
𝜇4𝑥−3𝑦+62
b. 𝜎4𝑥−3𝑦+62
x
y
Mean
54
87
St. Dev.
3
5