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Section D E
Test 3: Version A
Write all your answers on these pages. Calculators are permitted as are the charts that accompany
this test. There are 50 points on this test.
1. Four different experiments are done to estimate the mean ACT score of entering Calvin students.
Four different random samples are chosen from the population of entering Calvin students and these
confidence intervals for the mean are reported:
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(a) (25.8, 30.1) length 4.3
(b) (24.5, 26.5) length 2.0
(c) (25.0, 27.9) length 2.9
(d) (24.7, 28.6) length 3.9
Fill in the blanks below with the letter of the confidence interval above most likely to fit the description:
sample size 9, 90% confidence interval
D
sample size 9, 95% confidence interval
A
sample size 36, 90% confidence interval
B
sample size 36, 95% confidence interval
C
Smaller sample sizes and larger confidence make wider intervals.
2. Suppose that you are in the following hypothesis testing situation:
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null
H0 : µ = 0
alternate Ha : µ > 0
and the sample size is 10. Suppose that your t-value is 1.89. What is the best that we can say about
the P -value of this statistic? .025 < P < .05
3. We use the t-distribution to construct confidence intervals for µ.
(a) What kind of variable (categorical or quantitative) do we have in this situation? Quantitative
(b) What is µ in terms of the variable? The mean of that variable in the population.
(c) We use the t-distribution instead of the normal distribution to account for the fact that we don’t
know σ.
(d) To use the t-distribution to construct confidence intervals for the mean we must have a sample
that is a SRS.
(e) Even if our sample is chosen appropriately, the confidence intervals that we construct using the
t-distribution are approximate unless the population is normally distributed.
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4. Often people who are obese have high blood pressure and can decrease their blood pressure by losing
weight. However some weight loss programs do not seem to help reduce blood pressure. A researcher
tested a certain weight loss diet and claimed that the patients lost significant weight (p < .01) but did
not decrease blood pressure significantly (p = .32). The researcher evidently recorded two variables
for each of the subjects tested: W (the amount of weight lost), and B (the decrease in blood pressure).
(a) For the conclusion concerning weight loss, what is the parameter about which the researcher is
making an inference? Say this in both words and symbols.
The mean weight loss, µW
(b) For the conclusion concerning weight loss, what is the null hypothesis?
H0 : µW = 0.
(c) The claim that “patients lost significant weight” is sloppy. What is the researcher implying by
this and why is it wrong to imply that?
Statistical significance is not practical significance. He is implying that the patients lost a lot of
weight when the weight losses could have been quite small on average and still give a statistically
significant result.
(d) For the conclusion concerning blood pressure decrease, what is the null hypothesis?
mean blood pressure decrease is 0. Say µB = 0.
(e) Why does the researcher claim that p = .32 means that the program “did not decrease blood
pressure significantly”?
A P -value of .32 means that the blood pressure decrease could have happened by chance alone.
A P -value of .32 is not statistically significant at any reasonable level of significance.
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5. In the most recent Zogby Poll, 1201 likely voters were asked who they were going to vote for and 586
(49%) of them said they were going to vote for Obama.
(a) What is the population in this story? Describe this as accurately as you can from the account
that is given here.
Evidently, all voters likely to vote in the election on Tuesday.
(b) Construct a 95% confidence interval for the proportion of this population that will vote for
Obama.
s
586
± 1.96
1201
586
− 1201
)
1201
586
(1
1201
or the plus 4 interval.
(c) The poll announced that the margin of error for this poll was 3%. A typical reader might think
that this means that the true proportion of voters who will vote for Obama is surely within 3%
of 49%. Explain to that reader why that is an incorrect conclusion. Your explanation should
clearly show that you know what a confidence interval means.
The margin of error for the confidence interval is not sure. A confidence interval will not contain
the true value 5% of the time.
(d) It is probably not the case that Zogby achieved a sample that was as good as our theoretical
models call for. List two ways in which Zogby’s sample might be biased and give a plausible
argument explaining the direction that these might bias (for or against Obama) the results.
many answers are possible.
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6. An airline wants to know the average lifetime of a certain kind of airconditioning unit installed on
its Boeing 747 planes. It keeps careful records on 12 of them and finds the following lifetimes:
3 5 7 18 43 85 91 98 100 130 230 487
A Crunchit summary and boxplot of this data is given by
(a) Write a 95% confidence interval for the mean lifetime of this kind of airconditioning unit.
√
s
x̄ ± t∗ √ = 108 ± 2.201 13612
n
(b) The construction of such an interval relies on a certain assumption about the population of
airconditioning units from which these came. What is that assumption and does it seem likely
to be true in this case?
The population is normally distributed. Not likely - the sample is skewed right with a large
outlier – perhaps the population is similarly skewed.
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7. (A hypothesis test) A medical testing laboratory claims to have a test for a certain disease that has
a sensitivity of 95%. This means that if the person has the disease, the probability that the test is
positive is 0.95. Suppose that we decide to test the laboratory’s claim as we do not think that the
sensitivity could be that high. We test 260 individuals that we know have the disease and find that
the lab correctly identifies only 239 of these as positive. The relevant hypothesis test is performed
and the Crunchit output is provided.
(a) State precisely the null and alternate hypotheses in terms of a parameter. (You’ll notice that
these have been removed from the Crunchit output.)
null
H0 : p = .95
alternate Ha : p < .95
(b) What exactly does the P -value say about the medical laboratories claim? Be sure that I know
that you know what the P -value means by interpreting the exact numerical value above.
The P -value gives strong evidence against the null hypothesis as, if the null were true, we would
expect to see a sample proportion this small at most 1.1% of the time.
(c) State the result of the hypothesis test in a good sentence that uses the language of statistical
significance.
The sensitivity of the test is statistically significantly lower than 95% at the 5% level of significance.
(d) Given the outcome of 239 correct tests out of 260, would you reject the null hypothesis if you
were only willing to tolerate a Type I error rate of 5%? Why or why not?
Yes, rejecting the null for this outcome would result in only 1.1% Type I errors.
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