Download Test 9A

Practice Test 9
AP Statistics
Name:
Directions: Work on these sheets.
Part 1: Multiple Choice. Circle the letter corresponding to the best answer.
1. Following a dramatic drop of 500 points in the Dow Jones Industrial Average in September 1998, a
poll conducted for the Associated Press found that 92% of those polled said that a year from now their
family financial situation will be as good as it is today or better. The number 92% is a
(a) Statistic
(b) Sample
(c) Parameter
(d) Population
(e) None of the above. The answer is
.
2. In a large population, 46% of the households own VCRs. A simple random sample of 100 households
is to be contacted and the sample proportion computed. The mean of the sampling distribution of the
sample proportion is
(a) 46
(b) 0.46
(c) about 0.46, but not exactly 0.46
(d) 0.00248
(e) the answer cannot be computed from the information given.
3. If a population has a standard deviation  , then the standard deviation of the mean of 100 randomly
selected items from this population is
(a) 
(b) 100 
(c)  /10
(d)  /100
(e) 0.1
4. The distribution of values taken by a statistic in all possible samples of the same size from the same
population is
(a) the probability that the statistic is obtained.
(b) the population parameter.
(c) the variance of the values.
(d) the sampling distribution of the statistic.
(e) none of the above. The answer is
.
5. If a statistic used to estimate a parameter is such that the mean of its sampling distribution is equal
to the true value of the parameter being estimated, the statistic is said to be
(a) random
(b) biased
(c) a proportion
(d) unbiased
(e) none of the above. The answer is
.
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6. A simple random sample of 1000 Americans found that 61% were satisfied with the service
provided by the dealer from which they bought their car. A simple random sample of 1000
Canadians found that 58% were satisfied with the service provided by the dealer from which they
bought their car. The sampling variability associated with these statistics is
(a) exactly the same.
(b) smaller for the sample of Canadians because the population of Canada is smaller than that of
the United States, hence the sample is a larger proportion of the population.
(c) smaller for the sample of Canadians because the percent satisfied was smaller than that for the
Americans.
(d) larger for the Canadians because Canadian citizens are more widely dispersed throughout the
country than in the United States, hence they have more variable views.
(e) about the same.
7. The central limit theorem is important in statistics because it allows us to use the Normal
distribution to make inferences concerning the population mean:
(a) provided that the sample size is reasonably large (for any population).
(b) provided that the population is Normally distributed and the sample size is reasonably large.
(c) provided that the population is Normally distributed (for any sample size).
(d) provided that the population is Normally distributed and the population variance is known (for
any sample size).
(e) provided that the population size is reasonably large (whether the population distribution is
known or not).
8. A phone-in poll conducted by a newspaper reported that 73% of those who called in liked business
tycoon Donald Trump. The unknown true percent of American citizens who like Donald Trump is
a
(a) Statistic
(b) Sample
(c) Parameter
(d) Population
(e) None of the above. The answer is
.
9. The law of large numbers states that, as the number of observations drawn at random from a
population with finite mean  increases, the mean x of the observed values
(a) gets larger and larger.
(b) gets smaller and smaller.
(c) gets closer and closer to the population mean  .
(d) fluctuates steadily between one standard deviation above and one standard deviation below the
mean.
(e) varies randomly about  .
10. Suppose we are planning on taking an SRS of size n from a population. If we double the sample
size, then  x will be multiplied by
(a) 2
(b) 1
(e) none of these
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(c) 2
2
(d) 1/2
Test 9A
11. If a statistic used to estimate a parameter is such that the mean of its sampling distribution is
different from the true value of the parameter being estimated, the statistic is said to be
(a) Random
(b) Biased
(c) A proportion
(d) Unbiased
(e) None of the above. The answer is
.
12. A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales
tax from 5% to 6%, with the additional revenue going to education. Let p̂ denote the proportion
in the sample who say they support the increase. Suppose that 40% of all adults in Ohio support
the increase. The standard deviation of p̂ is
(a) 0.40
(b) 0.24
(c) 0.0126
(d) 0.00016
(e) 0
Part 2: Free Response
Communicate your thinking clearly and completely.
For Questions 8–10, consider the process of repeatedly taking SRSs of size n from a heavily leftskewed population distribution with mean µ and standard deviation  and calculating the sample
mean x .
13. What is the mean of the sampling distribution of x ?
14.
What is the standard deviation of the sampling distribution of x ?
15. What is the shape of the sampling distribution of x ? Justify your answer.
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16.
The following window shows the results of a computer simulation that is designed to illustrate
sampling distributions.
Briefly discuss the bias and variability of the sampling distribution of the sample median
17. Suppose that you and your lab partner flip a coin 20 times and you calculate the proportion of tails to
be 0.8. Your partner seems surprised at these results and suspects that the coin is not fair. Write a
brief statement that describes why you either agree or disagree with him. Your response should relate
to the principles in Chapter 9 of our text.
18. This problem extends the previous problem. You flip the coin 20 more times and calculate the
proportion of heads. You repeat the process again and again until you have calculated 25
proportions p̂ . Your lab partner then plots a histogram of the results on his calculator and overlays a
boxplot for the same data as shown. Your lab partner appears totally bewildered now.
Interpret these results for him.
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Test 9A
19. A study of college freshmen’s study habits found that the time (in hours) that college freshmen use
to study each week follows a distribution with a mean of 7.2 hours and a standard deviation of 5.3
hours.
(a) Can you calculate the probability that a randomly chosen freshman studies more than 9 hours?
If so, do it. If not, explain why not.
(b) What is the shape of the sampling distribution of the mean x for samples of 55 randomly
selected freshmen? Justify your answer.
(c) What are the mean and standard deviation for the average number of hours x spent studying
by an SRS of 55 freshmen?
(d) Find the probability that the average number of hours spent studying by an SRS of 55 students
is greater than 9 hours. Show your work.
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20. In a test for ESP (extrasensory perception), the experimenter looks at cards that are hidden from
the subject. Each card contains either a star, a circle, a wave, or a square. As the experimenter
looks at each of 20 cards in turn, the subject names the shape on the card.
(a) Describe the design of a simulation to estimate the proportion of correct guesses assuming the
subject does not have ESP using the partial random digits table below. Then perform the
simulation.
3 5 4 7 6
6 0 9 4 0
5 5 9 7 2
7 2 0 2 4
3 9 4 2 1
1 7 8 6 8
6 5 8 5 0
2 4 9 4 3
0 4 2 6 6
6 1 7 9 0
3 5 4 3 5
9 0 6 5 6
When the ESP study described above discovers a subject whose performance appears to be better
than guessing, the study continues at greater length. The experimenter looks at many cards bearing
one of five shapes (star, square, circle, wave, and cross) in an order determined by random
numbers. The subject cannot see the experimenter as he looks at each card in turn, in order to
avoid any possible nonverbal clues. The answers of a subject who does not have ESP should be
independent observations, each with probability 1/5 of success. We record 1000 attempts.
(b) What are the mean and standard deviation of the proportion of successes among the 1000
attempts?
(c) Find the probability that a subject without ESP will be successful in at least 24% of the 1000
attempts. Show your work.
(d) The researcher considers evidence of ESP to be a proportion of successes so large that there is
only probability 0.01 that a subject could do this well or better by guessing. What proportion
of successes must a subject have to meet this standard?
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Test 9A
21. A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax
from 5% to 6%, with the additional revenue going to education. Let p̂ denote the proportion in the
sample who say they support the increase. Suppose that 40% of all adults in Ohio support the
increase.
(a) If p̂ is the proportion of the sample who support the increase, what is the mean of the sampling
distribution of p̂ ?
(b) What is the standard deviation of the sampling distribution of p̂ ?
(c) Explain why you can use the formula for the standard deviation of p̂ in this setting.
(d) Check that you can use the Normal approximation for the distribution of p̂ .
(e) Find the probability that p̂ takes a value between 0.37 and 0.43.
(f) How large a sample would be needed to guarantee that the standard deviation of p̂ is no more
than 0.01? Explain.
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22. An opinion poll asks a sample of 500 adults (an SRS) whether they favor giving parents of schoolage children vouchers that can be exchanged for education at any public or private school of their
choice. Each school would be paid by the government on the basis of how many vouchers it
collected. Suppose that in fact 45% of the population favor this idea.
(a) What is the mean of the sampling distribution of p̂ , the proportion of adults in samples of 500
who favor giving parents of school-age children these vouchers?
(b) What is the standard deviation of p̂ ?
(c) Explain why you can use the formula for the standard deviation of p̂ in this setting.
(d) Check that you can use the Normal approximation for the distribution of p̂ .
(e) What is the probability that more than half of the sample are in favor? Show your work.
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23. A certain beverage company is suspected of underfilling its cans of soft drink. The company
advertises that its cans contain, on the average, 12 ounces of soda with standard deviation 0.4 ounce.
For the questions that follow, suppose that the company is telling the truth.
(a) Can you calculate the probability that a single randomly selected can contains 11.9 ounces or less?
If so, do it. If not, explain why you cannot.
(b) A quality control inspector measures the contents of an SRS of 50 cans of the company’s soda and
calculates the sample mean x . What are the mean and standard deviation of the sampling
distribution of x for samples of size n = 50?
(c) The inspector in part (b) obtains a sample mean of x  11.9 ounces. Calculate the probability that
a random sample of 50 cans produces a sample mean fill of 11.9 ounces or less. Be sure to
explain why you can use a Normal calculation.
(d) What would you conclude about whether the company is underfilling its cans of soda? Justify
your answer.
I pledge that I have neither given nor received aid on this test. __________________________
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Test 9A