### hw4 (Due 9/24)

```Stat 514
Due 9/24
Assignment #4
- Montgomery - Chapter 3
1. The following data are the treatment means from an experiment where each treatment was randomly
and equally allocated to a total of 27 experimental units.
Trt
Mean
1
3
2
1
3
0
Suppose the experimenter planned to test the following four hypotheses and assume M SE = 6.
a) H0 :
b) H0 :
c) H0 :
d) H0 :
3µ2 = µ3 + 2µ1
µ1 = µ3
µ1 + µ2 + µ3 = 9
µ1 + µ3 = 2µ2
a Using α = .05, test each of the four hypotheses (two sided).
b Which of the four linear combinations of means are contrasts? Why?
c Are any pairs of contrasts orthogonal? Which ones and why?
d Suppose the experimenter was also interested in testing H0 : µ1 = 0. Since the sample mean of
the third group is zero, it appears this is the same test as hypothesis b). Is it? Explain why
or why not?
2. Ten needles were randomly selected from a branch of a loblolly pine tree. The stomata (microscopic
breathing holes) are arranged in rows. On each needle, four rows are randomly selected and the
number of stomata per centimeter for each of the rows was determined. The data below is in the
file named stomata.dat.
Needle
1
2
3
4
5
6
7
8
9
10
149 136 143 121 148 129 127 134 117 129
143 139 142 133 121 134 130 137 128 132
138 129 124 126 124 127 123 119 117 131
131 143 134 130 128 113 125 130 118 137
a Why is the random effects model appropriate here?
b Estimate all relevant variance components.
c What percentage of the overall variation in stomata number per centimeter is due to the needle?
d Construct a 95% CI for this ratio.
e Compute a 95% confidence interval for the average number of stomata per centimeter.
3. A sociologist is interested in studying the IQs of teachers from low income areas of a major city. Six
schools were randomly chosen from low income areas and from each of these schools, five teachers
were randomly chosen. The following table summarizes the mean IQ for each of these schools
(NOTE: These numbers are all made up and are not intended to reflect teachers’ true IQ scores).
School
Mean
1
97
2
99
3
94
4
109
5
98
6
103
a If MSE =40, is there significant variability in average IQ among schools in low income areas (use
α = .01)?
b Estimate all variance components.
c How much power does this study have if the true variances were such that 2στ2 = σ 2 and n were
increased to 10?
d Suppose the national average IQ for teachers is 105. Test the null hypothesis that the average
IQ of these teachers is not lower than the national average (α = .05).
4. A random sample of 48 devices was selected from a warehouse and in turn randomly divided into
three groups of 16. Group A was the control group so nothing was done to the devices. Group
B devices were submerged in water for 15 minutes. Group C devices were dropped to a cement
floor from 5 feet. Each device was then evaluated for performance (the lower the score the better).
Does there appear to be any measurable effect of immersing them in water or dropping them?
Perform the appropriate analysis (F test and multiple comparison procedure). (NOTE: This data
set is available in the data directory (perform.dat))
A
0.38
0.25
0.43
0.33
0.39
0.51
0.75
0.51
0.53
0.43
0.34
0.51
0.35
0.33
0.42
0.33
B
0.53
0.33
0.38
0.45
1.09
0.56
0.48
0.55
0.75
0.37
0.44
0.48
0.53
0.69
0.41
0.44
C
0.48
0.65
0.44
0.47
0.43
0.43
0.41
0.75
0.58
0.38
0.48
0.41
0.62
0.53
0.39
0.49
5. A psychologist has tested 10 independent hypotheses. She has decided to control the false discovery
rate for this set of hypotheses at 0.05 . The ten p-values she has obtained are as follows: 0.04, 0.15,
0.02, 0.31, 0.06, 0.62, 0.01, 0.03, 0.46, 0.08 . Which, if any, hypotheses can she reject controlling
the FDR at 0.05? Which, if any, hypotheses can she reject controlling the experimentwise error
rate at 0.05?
2
6. You want to compare 3 treatments using a one-way fixed effects model. In designing your experiment
you decide you want at least 80% power (at the 5% significance level) if the treatment means were
as different as 100 - ∆, 100, and 100 + ∆. Suppose that ∆ = 5 and σ 2 = 10. How large must n
be?
7. Suppose that this experiment in #6 will use n = 4 observations from each group. Staying with 5%
significance, how small a difference can this experiment detect with 80% power?
3
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