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Statistics / Mr. Hansen |
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Name:
__________________________________ |
Test
on Chapters 9 and 10
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Instructions and Scoring.
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1.(a) |
Sketch a histogram of an
extremely skew left distribution that exhibits both gaps and outliers. |
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(b) |
Estimate the population mean
and population s.d. of your distribution. No work is required, but your
estimates should be reasonable. Use the correct one-letter identifiers. |
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(c) |
Assume that your population
contains many thousands of individuals, and consider the sampling distribution
of means drawn from your distribution using samples of size n = 150. Compute the mean and s.d. of
this sampling distribution, using
the correct notation that we learned. Only a small bit of work is needed. |
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(d) |
Provide a reason (complete sentence
not required) that justifies your steps in part (c). |
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2. |
A soft-drink bottling
company asserts that its bottling machine is designed to fill 16-oz. bottles
to a mean fill level of 16.2 fluid ounces, with s.d. of 0.3 fluid ounces. |
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(a) |
Assuming that the company
is telling the truth, compute the percentage of bottles that will be
underfilled, i.e., below the claimed fill level of 16 fl. oz. |
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(b) |
In a case (24 bottles)
pulled randomly off the packing line, what is the probability that the case
contains less than 384 fluid ounces of product? Again, assume that the
company is telling the truth. |
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(c) |
Explicitly state any other
assumption(s) you made in order to compute your answer to parts (a) and (b). |
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(d) |
Are the assumption(s) in
part (c) realistic? Is there cause to doubt the validity of someone’s numeric
answer to part (b)? |
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(e) |
A government agency samples
10,000 cases randomly and finds several that have less than 384 fluid ounces
of product. Is this sufficient evidence to indict the company on a conspiracy
of underfilling bottles? Explain briefly; no work required. |
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(f) |
A consumer group samples
one case, selected at random, and finds that it has less than 384 fluid ounces
of product. Is this strong evidence to doubt the accuracy of the company’s
assertion that the mean fill level is 16.2 fluid ounces? |
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(g) |
A different consumer group
samples 11 bottles and random and finds a mean fill level of 16.1 fluid
ounces. Based on this data set alone, plus the company’s s.d. claim of 0.3
fluid ounces, compute a 90% confidence interval for the parameter of
interest. Show work, and write your conclusion as a complete sentence. |
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3. |
Oral question: At some point during the test, I will ask you to speak with me
privately to describe what the law of large numbers says. I will also ask you
a specific question regarding the case of phat ® p. Here
is that question: Is it true that as n
® ¥, the expected difference between the expected count
of successes (np)
and the actual count of successes (observed count) approaches 0? In other
words, does the number of successes in n
trials approach np
as n ®
¥? |
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4. |
Essay:
Explain in several thoughtful sentences the reason(s) for a paradox of American
life. Most of the people who return from Las Vegas with reports of their
gambling experience volunteer that they won or broke even, yet casinos are
profitable. In fact, the probability of a gambler’s winning in Las Vegas is
fairly high, meaning that these returning tourists are not lying. |
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5. |
Explain briefly (no work
needed) the effect of decreasing n
on each of the following: |
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(a) |
The m.o.e. of a 95%
confidence interval. |
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(b) |
The P-value of a test, when H0
is false and all facts about the population remain unchanged as n decreases. |
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(c) |
The power of a fixed level a test, when a, the alternative hypothesis, and all facts about
the population remain unchanged. |
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6. |
[See #10.86 on p. 583 of
textbook. The question is copyrighted, and although re-using it on a test is
legitimate, posting it on the Web is not.] |