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).
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.
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.
c) Conduct one F-test of the null hypothesis H0 : The level of swine manure (none, low, or high) has no effect on mean yield vs. the alternative HA : 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.