Honors Introduction
to Statistics
Practice Problems for
Test 3
Show your work for full credit. Feel free to check your answers with your calculator, but answers without supporting work will receive little or no credit. Always interpret results in the context of the situation.
1.
Harley-Davidson motorcycles make up 14% of all
motorcycles registered in the
a. If Harley’s
make up 14% of motorcycles stolen, what would be the sampling distribution of
the proportion of Harleys in a sample of 9224 stolen motorcycles?
b. Is the
proportion of Harleys among stolen bikes significantly higher than their share
of all motorcycles? You could use a
hypothesis test to answer this question, but computations are really not
necessary. Explain why not.
2.
A college president says, “99% of the alumni support my
firing of Coach Boggs.”
a. Describe the
population and explain in words what the parameter p is.
b. You contact
an SRS of 200 of the college’s 15,000 living alumni and find that 152 of them
support firing the coach. Give the
numerical value of the statistic 
that estimates p.
c. Based on the responses of the alumni
you contacted, construct a 99% confidence interval for p. (Your work should exhibit a correct critical
value. Feel free to check your result
using your calculator, but an interval with no work will receive no credit.)
d. Explain the
meaning of your computation in Part (c).
How does your response relate to the president’s assertion?
3.
When we toss a coin and call heads or tails to make a
decision, we are generally assuming that coins are “fair,” that is, that there
are equal chances that a flipped coin will turn up heads and tails. What if, instead of flipping pennies, we tip
them? (Carefully set a penny on its edge
on a table or other sturdy but movable surface, then jar the table to make it
fall over.) Your friend claims that
pennies are more likely to turn up heads than tails when they are tipped, and
you decide to test her claim by performing a hypothesis test.
a. State your
hypotheses, both in symbols and in words.
b. Suppose you randomly choose 50
pennies, set them on their edges, and tip them.
Of the 50 pennies, 32 come up heads.
Decide if this is reason to believe your friend’s claim. (Compute the test statistic and the P-value,
and then clearly state your conclusion.)
4.
The carapace lengths (in mm) of 15 mature gopher
tortoises randomly selected from the preserve in Abacoa are shown below.
320 295 284 303 315 308 303 305
272 315 291 294 276 318 278
a. Examine these data
for shape, center, spread, and outliers.
b. Do you
believe the use of our inference techniques is justified in this
situation? Explain your answer.
c. Give a 95%
confidence interval for the mean carapace length of all mature gopher tortoises
in the preserve. Write a complete
sentence interpreting the meaning of your interval. (Your sentence should say
something about tortoises!).
d. Estimate the
sample size you would you need to compute a 95% confidence interval with a
margin of error less than 3 mm? Why
can’t you give an exact answer?
5.
A study of computer-assisted learning examined the
learning of “Blissymbols” by children.
The researcher designed two computer lessons that taught the same
content, one in which students interacted with the material, and one in which
students controlled the pace of the lesson but otherwise did not interact with
the program. After the lesson, the
computer presented a quiz that asked the children to identify 56
Blisssymbols. Here are the numbers of correct
identifications by the 24 children in the Active group:
|
29 |
28 |
24 |
31 |
15 |
24 |
27 |
23 |
20 |
22 |
23 |
21 |
|
24 |
35 |
21 |
24 |
44 |
28 |
17 |
21 |
21 |
20 |
28 |
16 |
And here are the
counts for the 24 children in the Passive group:
|
16 |
14 |
17 |
15 |
26 |
17 |
12 |
25 |
21 |
20 |
18 |
21 |
|
20 |
16 |
18 |
15 |
26 |
15 |
13 |
17 |
21 |
19 |
15 |
12 |
a. Is there good
evidence that active learning is superior to passive learning? State your hypotheses, give a test statistic
and P-value, and clearly state your conclusion in the context of student
learning.
b. Give a 90%
confidence interval for the difference in the mean number of Blissymbols
identified correctly by the active learning group and the passive learning
group. Interpret your result.
c. What
assumptions do your procedures from (a) and (b) require? Do the data meet these assumptions? Justify your answer.
6. Twelve
runners are asked to run a 10-kilometer race on each of two consecutive
weeks. In one of the races the runners
wear one brand of shoe and in the other a second brand. The brand they wear in each race is determined
at random. All runners are timed and are
asked to run their best in each race.
The results (in minutes) are given below.
|
Runner |
Brand 1 |
Brand 2 |
|
1 |
31.23 |
32.02 |
|
2 |
29.33 |
28.98 |
|
3 |
30.50 |
30.63 |
|
4 |
32.20 |
32.67 |
|
5 |
33.08 |
32.95 |
|
6 |
31.52 |
31.53 |
|
7 |
30.68 |
30.83 |
|
8 |
31.05 |
31.10 |
|
9 |
33.00 |
33.12 |
|
10 |
29.67 |
29.50 |
|
11 |
30.55 |
30.57 |
|
12 |
32.12 |
32.20 |
Use the
appropriate procedure to determine if there is evidence that the brand of the
shoe affects runners’ times. State your
hypotheses, compute the test statistic, give the P-value (or an estimate of
it), and interpret your result.
7.
The Physician’s Health Study examined the effects of
taking an aspirin every other day.
Earlier studies suggested that aspirin might reduce the risk of heart
attacks. The subjects were 22,071
healthy male physicians at least 40 years old.
The study assigned 11,037 of the subjects at random to take
aspirin. The others took a placebo. The study was double-blind. The researchers found that 119 participants
in the Aspirin group had strokes, while 98 of those in the Placebo group had
strokes. Is this difference significant? Conduct the appropriate test, be sure the
technical conditions for the test have been satisfied, and state your
conclusion.
8.
How do we estimate the standard deviation of the
sampling distribution when computing confidence intervals for the difference in
proportions? When conducting
significance tests to compare proportions from two populations? Explain why we use different things, and how
the two are related.
9. We
might be interested in the number of final exams that are canceled (including
ones given as a take-home or other alternate form). Is the frequency of departures from an
"in-class" final related to the subject area? Suppose that 45 courses are randomly selected
and the type of final exam in each is classified to give the two-way table below.
|
|
In-class |
Other |
|
Humanities |
6 |
11 |
|
Social Sciences |
9 |
6 |
|
Natural Sci/Math |
12 |
1 |
a. What sort of test
would you perform to answer the question "Is the frequency of departures
from an 'in-class' final related to the subject area?" State your hypotheses
b. Given that the value of the test statistic is 9.98, test to
determine if there is any relationship between the subject area of the course
and the type of final given. Estimate a
P-value and state your conclusions in a complete sentence (say something about
finals and subject areas).