Spring 2016
STATISTICS 402B
Midterm Exam I (100 points)
1. In a nutrition study, three supplements (A, B, and C) were compared by feeding month-old mice over a
period of 4 weeks. Twenty-four month-old laboratory mice, all alike in health and weight, were randomly
divided into four groups of 6 mice each. All twenty-four mice were weighed initially and mice of each group
were individually fed the 4 diets as follows over the next four-weeks:
Diet
1
2
3
4
Description
standard feed with no supplement
standard feed with supplement A
standard feed with supplement B
standard feed with supplement C
At the end of the 4-week period all twenty-four mice were weighed again and the weight gain is recorded in
milligrams. In animal experiments like this weight gain, among other measurements, is used as a measure
of growth. Management of the experiment (daily feeding, weighing etc.) were carried by the members of
the team of researchers and using the same set equipment, and the mice were housed in the same area of
the laboratory where the environmental conditions (such as heating, humidity, and light) were maintained
fairly constant.
(a) Why is this study an experiment and not an observational study?
(b) What is the response variable? Is this variable categorical or continuous-valued numerical?
(c) What is the factor under study? What are the treatments? Be specific.
(d) Identify an external source of variability that is controlled in the study and indicate how that variability is controlled.
(e) Is there a control/check in the study? Explain briefly.
(f) How is replication within this study accomplished?
(g) What is the degrees of freedom for the total sum of squares of the data from this study?
(h) In the analysis of variance of the data from this study, what is the error degrees of freedom?
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(i) Suppose during the experiment, one mice died in one group, and another two in another group. What
would be the answers to the two previous parts, in this case?
(j) (Bonus) Suggest an external source of variability that is NOT controlled in the study that might affect
growth rates of mice?
2. A municipality is considering two rivers (A and B) as supplementary sources for the city’s water supply.
To compare the pollution levels of the two rivers, 22 samples of water, 10 at random points from River A
and 12 at random points from River B, were selected and pollution levels (in ppm.) measured.
(a) If the investigators want to use a two-sample t-test to answer the question “Does this data indicate a
difference in pollution levels in the two rivers”? State three conditions that the two samples require
to satisfy in order to be able to use the t-test.
(b) Assume that µA and µB denote the means of the population of pollution levels of the two rivers,
respectively, state the hypotheses (null and the alternative) that you need to test in order to answer
the question in part a).
The following statistics were calculated from the data:
River
A
B
n
10
12
ȳi.
13.0
8.0
s2i
4.0
4.4545
(c) Assuming that the variances of the populations of pollution levels of the two rivers are the same,
calculate a pooled variance estimate of this common variance σ 2 . What is the degrees of freedom of
this estiamte?
(d) Calculate a t-statistic to test the hypothesis you stated in part b).
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(e) Use the attached t-table to test the above hypothesis (either by approximating a p-value from this
table or by determining a rejection region) for rejecting the null hypothesis.
3. A physical education researcher wished to study the effects of 4 teaching techniques on swimming profficiency for individuals in a beginning swimming course. A group of 24 nonswimming college women was
randomly divided into 4 groups of six subjects each. Instructions were provided to each group usinf one of
the 4 techniques, and at the end of the course, each subject was timed (in random order) while swimming
a distance of 10 yards.
Treatment
1. Verbal cues
2. Verbal + video
3. Video only
4. Control
Time (in
21.1 17.9
17.6 18.2
18.9 18.4
20.3 21.8
18.7
19.9
19.1
18.6
seconds)
19.5 20.6
20.0 16.9
18.8 17.7
19.7 20.9
18.3
17.5
20.5
20.8
(a) Identify the experimental units, treatments, and the response in this experiment.
(b) If the experimenters originally wanted to be able to detect a difference in treatment means
of 2 standard deviations (i.e., 2σ) with a power of 0.9 using a t-test at α = .05, what is the
total number of individuals that should have been used for the experiment?
The experiment was performed with a total of 24 individuals randomly assigned to the 4
treatment groups as above and the results shown above were analyzed using JMP. Refer to
the attached sheet titled Analysis of Swim Data for the JMP output.
(c) Report the ANOVA table to test the hypothesis that the treatment means are different
against the alternative that at least one is not different from others. Be sure to report the
appropriate test statistic and the p-value. State your decision, reason for the decision and
a conclusion within the context of the problem. Use α = 0.05.
(d) Compute the LSD for comparing treatment means at α = .05
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(e) Using the LSD connecting letters report from JMP, construct the LSD underscoring
procedure with means in increasing order here:
(f) Use the LSD procedure to indicate treatment levels that have significantly different treatment means and which don’t.
(g) Calculate the value for Tukey HSD for comparing treatment means at α = .05. Use the Q
value from the JMP output.
(h) Use the Tukey HSD connecting letters report from JMP to construct the HSD underscoring procedure with means in increasing order here:
(i) Use the HSD procedure to indicate treatment levels that have significantly different treatment means and which don’t. Are there any different conclusions from those of the LSD
procedure?
(j) What can you conclude from the two sets of plots constructed using JMP about your model
assumptions?
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