|
Biology 401
- Biometry
|
Fall
2003 |
Sample Questions for Exam 2 - Answers
Note - there are more questions in this sample than there
will be in the real exam
Test significance at alpha = 0.05 unless otherwise indicated.
- Using the data below, calculate the following confidence intervals.
Data: 6.4, 8.1, 5.9, 6.9
- The 95% CI
5.32 - 8.33
- The 99.9% CI
0.73 - 12.92
- The 95% CI if the sample size was increased by one but the mean and variance
stayed the same.
5.65 - 8.00
- You want to determine whether a certain hormone affects growth rates in
mammals. Using mice as a model organism, you take two siblings from each of
6 litters of mice, and in each case inject one sibling with the hormone and
the other with a placebo. You then weigh the mice two weeks later. The resulting
data are below.
|
Litter |
Hormone |
Placebo |
|
A |
3.6 |
5.2 |
|
B |
3.6 |
5.5 |
|
C |
4.4 |
4.0 |
|
D |
3.7 |
4.6 |
|
E |
4.8 |
5.3 |
|
F |
3.6 |
3.8 |
- Perform an appropriate test, being sure to show your work.
This is a paired t-test (or a repeated-measures ANOVA), because
the hormone and placebo values are not independent - they are paired because
of the relationship among the siblings.
P = 0.08
- What type of error (type I or II) might you have made above? Explain.
Could be a Type II error (false negative). Since the null hypothesis
was accepted, a Type I error (false positive) is not possible.
- What are the assumptions of the ANOVA test? That is, what conditions are
the data supposed to meet?
Random sample of independent values.
Populations are normally distributed.
Populations have homogeneous variances.
- You are investigating the general effects of smoking on physical activity
capacity. You round up 9 nonsmokers, 9 moderate smokers and 9 heavy smokers
as test subjects. You then choose three tests of activity capacity –
bicycle ergometer, treadmill and step – and assign three people from
each category to each test. The data obtained from the tests are the minutes
until the subject becomes exhausted (data below).
| |
Bicycle |
Treadmill |
Step |
| Nonsmokers |
12.8
13.5
11.2 |
16.2
18.1
17.8 |
22.6
19.3
18.9 |
Moderate
smokers |
10.9
11.1
9.8 |
15.5
13.8
16.2 |
20.1
21.0
15.9 |
Heavy
smokers |
8.7
9.2
7.5 |
14.7
13.2
8.1 |
16.2
16.1
17.8 |
- What type of test would you run on these data?
Two-factor ANOVA
- Is each factor fixed or random?
The "smoking" factor is definitely fixed. I would call the
activity measure random, since the three "levels" examined are
simply measures of activity capacity chosen from a larger set of similar
measures - we have no particular interest in those three tests. But an
argument for fixed could be made.
- What are the results of the statistical test you ran, and what do you
conclude from them?
ANOVA Table
| Source |
SS |
DF |
MS |
F |
P |
|
| Smoking |
84.899 |
2 |
42.449 |
60.230 |
0.0010 |
| Activity |
298.072 |
2 |
149.036 |
45.279 |
<0.0001 |
| Smoking * Activity |
2.815 |
4 |
.704 |
.214 |
0.9273 |
| Residual |
59.247 |
18 |
3.291 |
|
|
|
Both smoking and the type of activity had a significant effect on the
number of minutes the subject could sustain activity. There was no interaction
effect, indicating that the two effects are independent of one another.
- Four strains of lab mice are being bred for increased litter size. The litter
size of eight dams (mothers) of each breed is determined – see the sheet
“Mice” in the Excel file.
- Carry out an ANOVA comparing the four breeds using these data. Find
the resulting P value as closely as possible using Zar’s table B.4.
ANOVA Table
| Source |
SS |
DF |
MS |
F |
P |
|
| Among |
39.344 |
3 |
13.115 |
2.970 |
0.0488 |
| Within |
123.625 |
28 |
4.415 |
|
|
| Total |
162.969 |
31 |
|
|
|
|
- If appropriate given the results in “a,” run a suitable post-hoc
test to see which breeds differ in litter size. Use P = 0.05 as the critical
value.
| Comparison |
p |
Difference |
SE |
q |
qcrit |
P |
|
| 2 vs 3 |
4 |
2.875 |
0.743 |
3.870 |
3.845 |
P < 0.05 |
| 2 vs 4 |
3 |
2.000 |
0.743 |
2.692 |
3.486 |
N.S. |
| 2 vs 1 |
2 |
0.750 |
0.743 |
1.010 |
2.888 |
N.S. |
| 1 vs 3 |
3 |
2.125 |
0.743 |
2.860 |
3.486 |
N.S. |
| 1 vs 4 |
2 |
1.250 |
0.743 |
1.683 |
2.888 |
N.S. |
| 4 vs 3 |
2 |
0.875 |
0.743 |
1.178 |
2.888 |
N.S. |
|
- The good people at Kaplan commission a study to determine whether or not
their courses really help students do better on the MCAT. They obtain MCAT
scores from students who did and did not take their course at three different
universities (data in Excel worksheet). Conduct an appropriate analysis of
these data “by hand” (in Excel), summarize the results (below or
in Excel) and interpret your results (below).
This is definitely a model III ANOVA with Kaplan fixed and school random
ANOVA Table
| Source |
SS |
DF |
MS |
F |
P |
|
| Kaplan |
10.667 |
1 |
10.667 |
0.346 |
0.616 |
| School |
31.583 |
2 |
15.792 |
1.901 |
0.178 |
| Kaplan * School |
61.583 |
2 |
30.792 |
3.707 |
0.045 |
| Error |
149.500 |
18 |
8.306 |
|
|
|
Neither the Kaplan course nor the students' school had a significant overall
effect on MCAT scores. However, there was a significant interaction term,
suggesting that the Kaplan course had a different effect at different schools
- that is, it may have helped at some but hurt at others.
- In a study of effective yet legal stimulants, a group of 10 students is
each given one of four beverages to drink at midnight and then takes a word
recall test at 2:00 am. The process is repeated for four nights, with each
person receiving a different beverage each night in a random order. Analyze
the resulting data (see Excel worksheet) using a repeated measures ANOVA.
If difference among beverages are found, run post-hoc tests to determine which
beverages are different from which. Describe your findings below.
Two-factor ANOVA Table (without replication)
| Source |
SS |
DF |
MS |
F |
P |
|
| Beverage |
53.075 |
3 |
17.692 |
3.445 |
0.031 |
| Student |
142.025 |
9 |
15.781 |
3.072 |
0.011 |
| Residual |
138.675 |
27 |
5.136 |
|
|
|
SNK Test
| Rank |
Beverage |
Mean |
|
| 1 |
Mtn Dew |
14.9 |
| 2 |
Coffee |
13.5 |
| 3 |
Sprite |
12.4 |
| 4 |
Water |
11.9 |
|
|
|
|
|
|
|
| Comparison |
p |
Difference |
SE |
q |
qcrit |
P |
|
| 1 vs. 4 |
4 |
3 |
0.717
|
4.186 |
3.901 |
Yes |
| 1 vs. 3 |
3 |
2.5 |
0.717 |
3.488 |
3.532 |
No |
| 1 vs. 2 |
2 |
1.4 |
0.717 |
1.954 |
2.919 |
No |
| 2 vs. 4 |
3 |
1.6 |
0.717 |
2.233 |
3.532 |
No |
| 2 vs. 3 |
2 |
1.1 |
0.717 |
1.535 |
2.919 |
No |
| 3 vs. 4 |
2 |
0.5 |
0.717 |
0.698 |
2.919 |
No |
|
Both beverage choice and student have a significant effect on test score,
but the only pairwise difference for the beverages was between Mountain Dew
and water.