Quiz (Review 5-9)
1.
AP Statistics
Name:
You’re in college now, and you want to investigate the attitudes of students at your school toward the faculty’s commitment to
teaching. The Student Government will pay the costs of contacting about 500 students.
a. Specify the exact population for your study; for example, will you include part-time students?
b. Describe your sample design. Will you use a stratified sample?
c. Briefly discuss the practical difficulties that you anticipate; for example, how will you contact the students in your sample?
2.
A game of chance is based on spinning a 1–10 spinner like the one shown two times in succession. The player wins if the larger
of the two numbers is greater than 5.
a. What constitutes a single play of this game? What are the possible outcomes resulting in win or lose?
A single play is two consecutive spins.
First spin 0
1
2
3
4
5
6
7
Second
spin
0
lose
lose
lose
lose
lose
lose
win
win
1
lose
lose
lose
lose
lose
lose
win
win
2
lose
lose
lose
lose
lose
lose
win
win
3
lose
lose
lose
lose
lose
lose
win
win
4
lose
lose
lose
lose
lose
lose
win
win
5
lose
lose
lose
lose
lose
lose
win
win
6
win
win
win
win
win
win
win
win
7
win
win
win
win
win
win
win
win
8
win
win
win
win
win
win
win
win
9
win
win
win
win
win
win
win
win
b. Describe a correspondence between the random digits from a random digit table and
outcomes in the game.
Use two-digit numbers 00-99 where the first digit is the first spin and the second digit is the
second spin.
c. Describe a technique using the randInt command on the TI-83 to simulate the result of a
single play of the game.
select one random number from 0 to 99. Single digit numbers will have a zero added to the front.
Chapter 1
Quiz 1.1A
8
9
win
win
win
win
win
win
win
win
win
win
win
win
win
win
win
win
Win
Win
Win
Win
d. Use the random digit table or your TI-83 to simulate 20 repetitions. If you use the table, begin at line 140; if you use the
calculator, first enter 123rand to provide a seed. Report the proportion of times you win the game.
answers vary!
3.
4.
Suppose you toss a coin and roll a die.
a.
List the outcomes in the sample space.
b.
Find the probability of getting a head and an even number.
c.
Find the probability of getting 1 head.
d.
Find the probability of getting a 1, 2, or 3 on the die.
Suppose a person was having two surgeries performed at the same time.
If the chances of success for surgery A are 85%, and the chances of success for surgery B are 90%, what are the chances that both
will fail?
Chapter 1
Quiz 1.1A
5.
The probabilities that a customer selects 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15,
respectively.
a. Construct a probability distribution (table) for the data and draw a probability distribution histogram.
b. Find P(X > 3.5).
c. Find P(1.0 < X < 3.0).
d. Find P(X < 5).
e. Find the mean of the random variable, X.
f. Find the standard deviation of X.
6.
Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If the person gets a 7, he wins $5. The cost to
play the game is $3. Find the expected payout for the game.
Chapter 1
Quiz 1.1A
7.
A manufacturer produces a large number of toasters. From past experience, the manufacturer knows that approximately 2% are
defective. In a quality control procedure, we randomly select 20 toasters for testing. We want to determine the probability that no
more than one of these toasters is defective.
a. Show that this can be modeled by a binomial distribution for the random variable X.
b. Determine the probability that exactly one of the toasters is defective.
c. Find the probability that at most two of the toasters are defective. (Include enough details so that it can be understood how
you arrived at your answer.)
d. Find the mean and standard deviation for the random variable X in the toaster problem. First, define the random variable X.
8.
When a computerized generator is used to generate random digits, the probability that any particular digit in the set {0, 1, 2, . . . ,
9} is generated on any individual trial is 1/10 = 0.1. Suppose that we are generating digits one at a time and are interested in
tracking occurrences of the digit 0.
a. Determine the probability that the first 0 occurs as the fifth random digit generated.
b. How many random digits would you expect to have to generate in order to observe the first 0?
Chapter 1
Quiz 1.1A
c. Construct a probability distribution histogram for X = 1 through X = 5.
9.
An opinion poll asks, “Are you afraid to go outside at night within a mile of your home because of crime?” Suppose that the
proportion of all adults who would say “Yes” to this question is p = 0.4.
a. Use the partial table of random digits below to simulate the result of an SRS of 20 adults. Be sure to explain clearly which
digits you used to represent each of “Yes” and “No.” Write directly on or above the table so that I can follow the results of
your simulation. What proportion of your 20 responses were “Yes”?
6 8 4 1 7
3 5 0 1 3
1 5 5 2 9
7 2 7 6 5
8 5 0 8 9
5 7 0 6 7
8 2 7 3 9
5 7 8 9 0
2 0 8 0 7
4 7 5 1 1
8 1 6 7 6
5 5 3 0 0
6 0 9 4 0
7 2 0 2 4
1 7 8 6 8
2 4 9 4 3
6 1 7 9 0
9 0 6 5 6
3 6 0 0 9
1 9 3 6 5
1 5 4 1 2
3 9 6 3 8
8 5 4 5 3
4 6 8 1 6
3 8 4 4 8
4 8 7 8 9
1 8 3 3 8
2 4 6 9 7
3 9 3 6 4
4 2 0 0 6
b.
Repeat the previous problem using the next consecutive lines of the digits table with one line per SRS until you have
simulated the results of 5 SRSs of size 20 from the same population. Compute the proportion of “Yes” responses in each
sample. These are the values of the statistic p̂ in 5 samples. Find the mean of your 5 values of p̂ . Is it close to p?
c.
The sampling distribution of p̂ is the distribution of p̂ from all possible SRSs of size 20 from this population. What
would be the mean of this distribution?
Chapter 1
Quiz 1.1A
d.
If the population proportion changed to p = 0.5, what would then be the mean of the sampling distribution?
10. Choose an SRS of size n from a large population with population proportion p having some characteristic of interest. Let p̂ be
the proportion of the sample having that characteristic.
a.
What is the mean of the sampling distribution?
b.
What is the standard deviation of the sampling distribution?
c.
Under what conditions can the formula for the standard deviation of p̂ be reasonably accurate?
d.
The sampling distribution of p̂ is approximately normal when the sample size n is “large.” According to our adopted rule
of thumb, what conditions do we require in order to use the normal approximation to the sampling distribution of p̂ ?
11. According to government data, 22% of American children under the age of 6 live in households with incomes less than the
official poverty level. A study of learning in early childhood chooses an SRS of 300 children. What is the probability that more
than 20% of the sample are from poverty households? (Remember to check that you can use the normal approximation.)
Chapter 1
Quiz 1.1A
12. The weights of newborn children in the United States vary according to the normal distribution with mean 7.5 pounds and
standard deviation 1.25 pounds. The government classifies a newborn as having low birth weight if the weight is less than 5.5
pounds.
a. What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth?
You choose three babies at random and compute their mean weight, x .
b. What are the mean and standard deviation of the mean weight x of the three babies?
c.
What is the probability that their average birth weight is less than 5.5 pounds?
d.
Would your answers to a, b, or c be affected if the distribution of birth weights in the population were distinctly nonnormal?
Chapter 1
Quiz 1.1A