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Assignment 2
NAME
1. In an experiment to measure the effectiveness of carbon tetrachloride as a worm killer, each of
10 rats was injected with 500 larvae of the worm, Nippostrongylus muris. Eight days later 5 of
the rats, randomly selected, each received a 0.126 cm3 solution of carbon tetrachloride. Two days
later the rats were killed and the numbers of adult worms were counted. These numbers were 378,
275, 412, 265, and 286 for the control rats and 123, 143, 192, 240, and 259 for the rats treated
with carbon tetrachloride.
(a) A 95% confidence interval for the difference in mean numbers of surviving worms between
the treated and control groups is
lower limit =
upper limit =
(b) Describe the probability model for variation in worm counts that was implicitly assumed to
be a good approximate model in constructing the confidence interval in part (a).
(c) What conclusions can be reached about the effectiveness of carbon tetrachloride as a
worm-killer from the results in part (a).
2. Assuming that the probability of the birth of a male child is 0.51 and the sex of one child in a
family is independent of the sex of any other child, what is the probability that in a family with 5
children
(a) at least one child is a girl?
(b) exactly 3 children are boys?
(c) the first born child is a boy?
(d) there is at least one boy and one girl?
(e) there all girls?
3. A sample of 50 households was randomly selected from the homeowners in Ames, Iowa and an
independent sample of 100 households was randomly selected from the homeowners in Des
Moines, Iowa. For each household a yes or no response was obtained to improve the quality of
primary and secondary education in the city? The responses are given below (artificial data).
City
Ames
Des Moines
Response
Yes
23
38
to
No, or not sure
27
62
question
(a) Construct a 95% confidence interval for the proportion of homeowners in Ames who would
support the tax increase.
lower limit =
upper limit =
(b) Perform the chi-square test of the null hypothesis that the proportions of homeowners who
would support the tax increase are the same in the two cities. Report:
X2 =
d.f. =
p-value =
State your conclusion
(c) Construct a 95% confidence interval for the difference in the proportions of homeowners
willing to support the tax increase.
lower limit =
upper limit =
(d) Suppose a larger sample will be taken in Des Moines. Estimate how large the sample size
must be so that the width of the 95% confidence interval for the proportion of homeowners
willing to support the tax increase does not exceed 0.02.
sample size =
4. In examing the effects of two sprays, denoted by A and B, on the control of codling moth injury
to apple tree was sprayed with spray A and a different apple tree in the same orchard was sprayed
with spray B. The total number of apples and the number of wormy apples were counted for each
tree. The results are:
Spray A: of 2130 apples, 1299 were wormy
Spray B: of 2190 apples, 1183 were wormy
(a) Compute the observed proportion of wormy apples for each spray.
Spray A
Spray B
(b) Compute the value of the chi-square test statistic for the null hypothesis that the sprays are
equally effective.
X2 =
d.f. =
(c) Do you think it is appropriate to compare the X2 value in part (b) to the percentiles of
chi-square distribution with 1 d.f. to obtain a p-value? Is the evidence provided by this
experiment sufficient for you to conclude that the sprays are not equally effective? Explain.
5. The following experiment was performed to compare two media for growing a certain bacteria
were obtained from 40 subjects diagnosed as having a certain disease. One of the samples was
assigned to be grown using medium A and the other with medium B. The number of cases in
which the bacteria grew were recorded. The results are given below.
Medium B
Grew
Did not grow
Medium A
Grew
30
6
Did not grow
2
2
(a) Using the continutity correction, compute the value of the chi-square statistic for testing the
null hypothesis that the media are the same with respect to the chance that the bacteria grows
X2 =
d.f. =
p-value =
(b) Construct a 95% confidence interval for the difference PA - PB, where PA and PB are the true
probabilities that the bacteria will grow with median A and B, respectively.
lower limit =
upper limit =
(c) Suppose the experiment was performed a different way. Suppose the 40 subjects had been
randomly divided into two groups of 20. One bacteria sample was taken from each subject.
The 20 samples in one group were grown with medium A and medium B was used for the 20
samples from the other group. Estimate the standard error for the difference P̂A - P̂B for this
experiment.
Standard error for P̂A - P̂B =
How much larger is this standard error than the one used in part (b)?
(d) Redo part (c) with 80 subjects divided into two groups of 40.
Standard error for P̂A - P̂B =
(e) Which experiment would be the best one to do?
6. The growth rates of two species of trees, denoted as A and B, were compared in the following
experiment. Ten different locations were selected for growing the trees. The available land area
in each location was divided into two parts, or plots, of equal size and Species A was randomly
assigned to be grown in one of the plots while Species B was planted in the adjacent plot. The
average height of the trees in each plot was measured 6 years after the seedlings were planted.
The results (in meters) were as follows:
Location
1
2
3
Species A
3.2
2.7
3.0
Species B
2.8
2.7
2.0
4
5
6
2.7
1.7
3.3
3.0
2.1
4.0
7
8
9
10
2.7
2.6
2.9
3.3
1.5
2.2
2.7
2.5
(a) Construct a 95% confidence interval for the difference in the mean heights of Species A and B.
lower limit =
upper limit =
(b) Compute the value of the t-test for the null hypothesis that the mean heights are equal for
Species A and B. Report:
t-statistic =
d.f. =
p-value =
What conclusion did you reach from this test?
(c) Estimate how many locations would be needed so that the width of a 95% confidence interval
for the difference in the mean heights would be about 0.04 meters
number of locations =
(d) Does it matter how the locations are selected? Explain.
(e) Would it matter if the size of the plots (or the number of trees planted) varied from one location
to the next? Explain.