**Stat 401AB
Exam 2
November 4, 2002**

** **

1. Researchers
are interested in finding genes in *Arabidopsis* plants that change their
level of messenger RNA (mRNA) production when exposed to ultraviolet light. Ten plants were randomly divided into
two groups of 5 plants each. Each
plant in one group was exposed to ultraviolet light for 12 hours. Each plant in the other group was
exposed to normal light for 12 hours.
The amount of mRNA produced by a particular gene (call it gene X) was
measured in each of the 10
plants. Below are summary
statistics for the log-transformed data.

**Light Treatment Average of 5 log mRNA Amounts Standard Deviation of 5 log mRNA Amounts
**

Normal
4.7
0.9

Ultra-violet
6.9
1.1

a) Is there convincing evidence that gene X
changes its mRNA production in response to ultraviolet light? Compute a test statistic, two-sided
p-value, and provide a brief conclusion that answers this
question.

b) Fill in the blank in the following
sentence with an appropriate estimate and more than one
word.

It
is estimated that the amount of mRNA produced by plants exposed to ultraviolet
light was

____________________________________
the amount of mRNA produced by plants exposed to normal
light.

c) Provide a 95% confidence interval to
accompany the estimate that you gave in part (b).

1.
(continued)

d)
Suppose that gene X was one of 50 genes tested by the researchers in the
experiment. Suppose the purpose of
the experiment was to scan all 50 genes in hopes of finding some that change
mRNA production. Use the Bonferroni
method to adjust the p-value for gene X computed in part (a) so that the method
of analysis will control the probability of one or more mistaken rejections of a
true null hypothesis across the 50 tests.

e)
Provide a brief conclusion about gene X to accompany the adjusted p-value that
you computed above.

2. There is a relationship between the pH
level in postmortem muscle of a steer carcass and the time that has passed since
the steer was slaughtered. To
investigate this relationship, 10 steer carcasses were assigned to be measured
for pH at one of five times after slaughter. The data along with some summary
statistics are provided below.

**Steer Time after Slaughter (hours)
pH**

1
1
7.02

2
1
6.93

3
2
6.42

4
2
6.51

5
3
6.07

6
3
5.99

7
4
5.59

8
4
5.80

9 5
5.51

10
5
5.36

--------------------------------------------

Mean
3.00
6.12

Stand.
Dev.
1.49
0.58

a)
Without doing any computations, circle the value that is closest to the sample
linear correlation coefficient between pH and time after
slaughter.

-0.984
-0.147
0.000
0.147
0.984

b)
Find the equation of the least-squares regression line that can be used to
describe the relationship between pH and time after
slaughter.

c)
Predict the pH level in the postmortem muscle of a steer 2.5 hours after
slaughter.

3. A team of researchers is studying
erosion on highway construction sites.
The banks of a freeway overpass can be quite steep. If it rains hard on a recently built
bank, soil can wash down and contribute to pollution in nearby streams. The researchers would like to compare
the effectiveness of two different grasses (A and B) at preventing erosion. Eleven new freeway banks are available
for study. The two grasses are
randomly assigned to two test plots on each freeway bank. Six months after planting, a rainfall
simulator is used to produce heavy artificial rainfall on the two test plots at
each freeway bank. The amount of
soil washed off each test plot following the artificial rain is recorded. The data on the amount of soil washed
off for each bank and grass type are provided in the table below. Units are not specified, but larger
numbers mean more soil is washed off.

**Freeway Bank 1 2 3 4 5 6 7 8 9 10 11
**

**Grass A**
32 11 47 7 26 18 28 39 36 55 34

**Grass B
**19 13 21 8 20 15 21 30 42 25 29

a) Use the Wilcoxon signed-rank test to
test for a difference between the two grass types regarding their ability to
prevent soil runoff. Provide a test
statistic and use a normal approximation to determine an approximate two-sided
p-value. State a brief
conclusion.

b) Determine an exact two-sided p-value for
the Wilcoxon signed-rank test conducted in part (a).

4. Commercial fertilizer contains nitrogen
that helps to increase corn yield.
Swine manure also contains high amounts of nitrogen. Much corn is grown each year in
Iowa. Also large amounts of swine
manure are produced each year in Iowa.
Thus it makes sense to examine the effectiveness of using swine manure as
a fertilizer for corn. Researchers
conducted an experiment to evaluate the effectiveness of applying swine manure
to soil both with and without applying commercial fertilizer. Three levels of manure were considered
(none, low, high). Two levels of
commercial fertilizer were considered (absent and present). A field was divided into 30 plots of
land. Five plots were randomly
assigned to each of 6 treatments in a completely randomized design. The six treatments can be described as
follows.

**Treatment
Description**

1 no
manure and no commercial fertilizer

2
low manure and no commercial fertilizer

3
high manure and no commercial fertilizer

4
no manure with commercial fertilizer (commercial fertilizer
only)

5
low manure with commercial fertilizer

6
high manure with commercial fertilizer

Plots
were treated in the spring and yields were recorded during the fall
harvest. A summary of the grain
yield data (in Mg/ha) is provided below.

**Treatment
** **1 2 3 4 5
6**

**Number of plots ** 5 5 5 5 5 5

**Average
** 8.3 11.3 14.0 11.0 13.9 14.1

**Variance ** 2.6 3.3 3.1 3.3 2.9 2.8

a) Provide a one-way ANOVA table for this
data. Include SOURCE, DF, SS, MS,
and F columns in your table.

b) Provide a p-value for the F-statistic
associated with your ANOVA table, and explain briefly what this p-value means in
the context of this data.

4.
(continued)

c)
Conduct one F-test of the null hypothesis *H*_{0
}*:
The level of swine manure (none, low, or high) has no effect on mean
yield*
vs. the alternative *H*_{A
}*:
The level of swine manure (none, low, or high) has some effect on mean
yield. *Give
the value of the test statistic, its degrees of freedom, and an approximate
p-value. Also state a brief
conclusion.