Essential Statistics
1.
Proportions Review
Name: __________________________
Do you drink the Cereal Milk? A USA today poll asked a random sample of 1012 US adults what
they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 67%
said they drink it. Suppose that 70% of US adults actually drink the milk.
a. Find the mean and standard deviation of the sample proportion that says they drink the
cereal milk.
b. Check that you can use the normal approximation for the distribution of 𝑝̂ .
c. Find the probability of obtaining a sample of 1012 adults in which 67% or fewer say they
drink the cereal milk. So you have doubts about the results of this poll?
d. What sample size would be required to reduce the standard deviation of the sample
proportion to one-half the value you found in (a)?
2. Voter registration records show that 68% of all voters in Indianapolis are registered as Republics.
To test whether the numbers dialed by a random digit dialing device really are random, you use
the device to call 150 randomly chosen residents telephone numbers in Indianapolis. Of the
registered voters contact, 73% are registered Republicans.
a. Is each of the boldface numbers a parameter or a statistics? Give the appropriate
notation for each.
b. What are the mean and standard deviation of the sample proportion of registered
Republications in samples of size 150 from Indianapolis?
c. Find the probability of obtaining an SRS of size 150 from the population of Indianapolis
Voters in which 73% or more are registered Republicans. How well your random digit
device working?
3. Each of the following situations requires a significance test about a population mean. State the
appropriate null hypothesis Ho and alternative hypothesis Ha in each case. Then, determine if
the claim is valid.
a. Each soccer ball produced by Brine has at most a 10% chance of being defective. The
soccer corner decides to test the theory by randomly selecting 54 balls from their recent
delivery and testing them. Out of the 54 balls test, 2 were defective.
b. Texas A&M University gives students credit for French class if they score above a 4 on
the AP exam. The AP board claims that at least 53% of students make above a 4 on the
French test. A&M decides to test this theory, and randomly selects 412 incoming
freshman who applied to the University and took the AP French Exam. Of the 412
students, 221 students made above a 4 on the test. Is this a valid claim?
4. The manager of an orchard expects about 70% of his apples to exceed the weight
requirement for "Grade A" designation. At least how many apples must he sample to be
90% confident of estimating the true proportion within ± 4%?
5. A statistics professor wants to see if more than 80% of her students enjoyed taking her
class. At the end of the term, she takes a random sample of students from her large class
and asks, in an anonymous survey, if the students enjoyed taking her class. Write the
hypothesis statements for this situation.
6. A recent Gallup poll found that 28% of U.S. teens aged 13-17 have a computer with
Internet access in their rooms. The poll was based on a random sample of 1028 teens and
reported a margin of error of ±3%. What level of confidence did Gallup use for this poll?
7. One month the actual unemployment rate in France was 13.4%. If during that month you
took a survey of 100 Frenchmen and constructed a confidence interval estimate of the
unemployment rate, which of the following would be true?
I.
The center of the interval was .134
II.
The interval contained .134
III.
A 99% confidence interval contained .134
8. In general, how does doubling the sample size change the confidence interval size?
9. Under what conditions would it be meaningful to construct a confidence interval estimate
when the data consists of the entire population?
10. Two 95% confidence interval estimates are obtained: I (.128,.272) and II (.138, .262). If
the sample proportions ( p̂ ) are the same, which has the larger sample size?
a)
b)
c)
d)
Interval I has a larger sample size
Interval II has a larger sample size
The sample sizes are the same
More information is needed to answer this question
11. A telephone survey of 400 registered voters showed that 256 had not yet made up their
minds 1 month before the election. At what confidence level can we say that between
60% and 68% of the electorate were still undecided at that time?
12. When an online news magazine asked viewers to click their agreement or disagreement,
300 out of 1200 respondents agreed with a statement that the most practical way of
becoming a millionaire is winning a lottery. Immediate feedback stated that 25% of the
viewers, with a margin of error of  2.5%, agreed with the statement. Fine print claimed
95% confidence. What is the proper conclusion?
a)
b)
c)
d)
e)
We are 95% confident that the proportion of viewers who believe the most
practical way of becoming a millionaire is winning a lottery is between .225 and
.275.
Without knowing whether both np and n(1 – p) are > 10, the calculation is
inappropriate.
Without knowing whether or not the 1200 respondents are 10% of all viewers, the
calculation is inappropriate.
The z-distribution was inappropriately used.
The data was not a SRS, so the calculation is inappropriate
13. There are 50,000 high school students in an extended metropolitan region. As each of
their students came in to register for classes, guidance counselors were instructed to use a
calculator to pick a random number between 1 and 100. If the number 50 was picked, the
student was included in a survey. For one of the many survey questions, 30% of the
students said they couldn’t live without instant messaging. Are the conditions met for
constructing a confidence interval of the proportion of this region’s teenagers who
believe they couldn’t live without instant messaging?
a)
b)
c)
d)
e)
No, there is no guarantee that a representative random sample is chosen.
No, the sample size is not less than 10% of the population.
No, np and n(1 – p) are not both greater than 10.
No, there is not reason to assume that the population has a normal distribution.
Yes, all conditions are met, and a confidence interval can be constructed
14. For a given large sample size, which of the following gives the smallest margin of error
in calculating a confidence interval for a population proportion?
a)
b)
c)
d)
e)
90% confidence with
95% confidence with
99% confidence with
90% confidence with
95% confidence with
pˆ =.15
pˆ =.15
pˆ =.15
pˆ =.23
pˆ =.23
15. If 64% of a sample of 550 shoppers leaving a shopping mall claim to have spent over
$25,
a. determine a 99% confidence interval for the proportion of all shoppers who spend
over $25.
b. Shopping mall management claims that 75% of all shoppers spend over $25 at
their mall per
trip. What does your confidence interval say about
this claim?
16. A research study gives a 95% confidence interval for the proportion of subjects helped by
a new anti-inflammatory drug as (0.56, 0.65).
a. Interpret this interval in the context of the problem.
b. What is the meaning of "95%" confidence interval as stated in the problem?