STAT 201: Exam 3 Review Please read!

STAT 201: Exam 3 Review
Please read!
• The third exam will be 30-40% multiple choice and 60-70% long and short answer type problems.
• The multiple choice questions are concept oriented. A good way to study for these is to read the book
and know your concepts.
• The long and short type problems are more computation and interpretive in nature.
• The list of concepts provided below is not comprehensive.
• Your solutions to problems on the exam should be neatly presented, have all work shown, have pictures
drawn where appropriate and must be in complete sentences in order to receive complete credit.
• All standard error formulae and test statistic formulae will be given to you. You will have to know
which formula is to be applied to which situation. A SAMPLE formula sheet is attached for your review.
Concept list
• Sampling distribution results for proportions
• Sampling distribution results for means
• Calculating probabilities for the sample mean(x̄) and the sample proportion of successes (p̂) based on
the sampling distribution.
• Constructing and interpreting confidence intervals for population proportions.
• Interpreting the confidence level of a confidence interval.
• Hypothesis tests for one sample proportions
• Confidence interval and hypothesis test for two proportions (independent samples)
• Making decisions and conclusions for hypothesis tests based on confidence intervals.
• Deciding whether parameter estimates will be within a confidence interval based on the results of a
hypothesis test.
• Identifying Type I and Type II errors in hypothesis tests.
• Confidence interval and hypothesis test for one sample means
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Problem list
1. A wine importer has the opportunity to purchase a large consignment of 1947 Chateau Lafite Rothschild
wine. Because of the age of the wine, some of it may have turned into vinegar. However, the only way
to determine whether a bottle is still good is to open it and drink some. As a result, the importer has
arranged with the seller to randomly select and open 50 bottles. Suppose the importer believes that 30%
of the bottles will be spoiled.
(a) What is the distribution of the sample proportion of spoiled bottles of wine?
(b) Find the probability that the sample proportion of spoiled bottles of wine exceeds 0.40?
(c) Find the probability that the sample proportion of spoiled bottles of wine falls between 0.25 and
0.42?
2. Suppose that the number of blemishes in a lot of casual wear has a distribution with mean 3 and standard
deviation 1.2. The distribution of the number of blemishes is not Normal as it deals with count data. A
quality control technician randomly samples 100 items in a lot. Answer the following questions...
(a) What are the mean and standard deviation of the sample mean x̄?
(b) What is the shape of the distribution of x̄ ? What guarantees you this fact?
(c) What is the probability that the mean number of blemishes in a sample of size 100 is at most 3.25?
3. At a particular university, students are sick and tired of taking final exams on Saturdays. So, they decide
to protest. Sadly, the Provost will only listen to your complaint if you can prove that more than 85% of
the student population is unhappy with Saturday finals. In a random sample of 1600 students, you find
that 1380 would like to eliminate Saturday finals.
(a) What would be the null and alternative hypotheses for this problem?
(b) Compute the value of the test statistic and the p-value.
(c) At the 0.10 level, would we reject or fail to reject H0 ?
(d) Based on your decision, do you think the Provost will listen to the students?
4. A recent Gallup poll asked 1006 Americans, “During the past year, how many books, either hardcover
or paperback, did you read either all or part of the way through?” Results of the survey indicated that
x̄ = 18.8 books and s = 19.8 books. Construct and interpret a 99% confidence interval for the mean
number of books American read either all or part of during the preceding year.
One Proportion CI and Hypothesis Test ¡
5. In a random survey of 226 college students, 20 reported being “only” children, i.e with no siblings.
(a) Construct and interpret a 95% confidence interval for the proportion of college students nationwide
that are “only” children.
(b) Is there significant evidence that the true proportion of college students nationwide that are “only”
children is less than 10%? Test this question.
6. You are interested in purchasing a house in a new subdivision and are interested in the average size of
the homes in that subdivision. In a random sample of 25 homes, you find a sample mean of 2127.94
square feet and a standard deviation of 437.276 square feet.
(a) Assuming that the sizes of homes are normally distributed, calculate and interpret a 95% confidence
interval for the true mean size of homes in this subdivision.
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(b) A realtor claims that the average size of homes in this subdivision is 2400 square feet. You believe
that the average size of homes in this subdivision is less than the realtor’s claim. Test your claim
at the 0.10 level.
7. In October 2010, the Gallup organization surveyed 1134 American adults and found that 441 had a gun
in the home. In October 2009, the Gallup organization had surveyed 1134 American adults and found
that 458 had a gun in the home. Suppose a newspaper article has a headline that reads, “Fewer Guns
in the Home.” Is this an accurate claim? Test the newspaper’s claim at the 0.05 level.
8. Students are reluctant to report cheating by other students. A student project put this question to a
SRS of 172 undergraduates at one university : “You witness two students cheating on a quiz. Do you go
to the professor?” Only 19 answered “Yes”.
(a) Construct and interpret a 95% confidence interval for the proportion of all undergraduates who
would report cheating.
(b) Also test the hypothesis that this proportion is less than 0.25. Use α = 0.05.
9. In 2002, the mean age of an inmate on death row was 40.7 years, according to data obtained from the
U.S. Department of Justice. A sociologist wondered whether the mean age of a death-row inmate has
changed since then. She randomly selects 32 death-row inmates and finds their mean age is 38.9, with a
standard deviation of 9.6. Test the sociologist’s claim at the 0.05 level.
10. The PACE project at the University of Wisconsin in Madison deals with problems associated with highrisk drinking on college campuses. Based on random samples, the study states that the percentage of
UW students who reported bingeing at least three times within the past two weeks was 42.2% (n=334)
in 1999 and 21.2% (n=843) in 2009. Compute and interpret a 95% confidence interval for the difference
in the proportion of students who binge at least three times within the past two weeks between the years
of 1999 and 2009.
11. The Harris Poll conducted a survey in which they asked, “How many tattoos do you currently have on
your body?” Of the 1205 males surveyed, 181 responded that they had at least one tattoo. Of the
1097 females surveyed, 143 responded that they had at least one tattoo. Construct and interpret a 99%
confidence interval for the difference in the proportion of males and females that have at least one tattoo.
12. Proposal A is on the ballot for the upcoming election. The following table cross classifies the favoring of
Proposal A with political party preference.
Democrats
Republicans
Total
Favor A
28
37
65
Oppose A
22
13
35
Total
50
50
100
Does support for Proposal A differ between Democrats and Republicans?
(a) What would be the null and alternative hypotheses for this problem?
(b) Compute the value of the test statistic and the p-value.
(c) At the 0.10 level, would we reject or fail to reject H0 ?
(d) Write a conclusion in the terms of the problem.
(e) Based only on your decision, would a 90% confidence interval for pD − pR contain 0? Explain.
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13. The lifetime of a certain type of automobile tire (in miles) is normally distributed with mean µ = 40, 000
and standard deviation σ = 5, 000.
(a) What is the sampling distribution of the mean lifetime of a random sample of 25 tires?
(b) What is the probability that the mean lifetime of 25 randomly selected tires of this type will exceed
than 42,000 miles?
14. According to a study done by Nick Wilson of Otago University Wellington, the probability that a
randomly selected individual will not cover his or her mouth when sneezing is 0.267.
(a) What is the sampling distribution of the sample proportion of people who will not cover their mouth
when sneezing for a random sample of 100 individuals.
(b) If we take a random sample of 100 individuals, what is the probability that the sample proportion
of people who will not cover their mouth when sneezing is between 0.2 and 0.3?
Exam 3 Formulae:
Z=
x−µ
SE
Z=
pb − p
SE
Zc =
pb − p0
SE
tc =
x̄ − µ0
SE
pb1 − pb2 − pnull
SE
σ
SE = √
n
s
SE = √
n
Zc =
s
p(1 − p)
n
s
p0 (1 − p0 )
n
s
pb(1 − pb)
n
s
pb1 (1 − pb1 ) pb2 (1 − pb2 )
+
n1
n2
SE =
SE =
SE =
SE =
s
SE =
pb(1 − pb)
1
1
+
n1 n2