9.6. A Regression Problem 5399.5.8.Given the following observations in a two-way classification witha=3,
b=4,andc= 2, compute theF-statistics used to test that all interactions are
equal to zero (γij= 0), all column means are equal (βj=0),andallrowmeans
are equal (αi= 0), respectively. Data are in the formxijk,i,jin the data set
sec951.rda.
Row/Column 1 2 3 4
1 3.1 4.2 2.7 4.9
2.9 4.9 3.2 4.5
2 2.7 2.9 1.8 3.0
2.9 2.3 2.4 3.7
3 4.0 4.6 3.0 3.9
4.4 5.0 2.5 4.29.5.9.For the additive model (9.5.1), show that the mean profile plots are parallel.
The sample mean profile plots are given by plottingXij·versusj,foreachi.These
offer a graphical diagnostic for interaction detection. Obtain these plots for the last
exercise.
9.5.10.We wish to compare compressive strengths of concrete corresponding to
a= 3 different drying methods (treatments). Concrete is mixed in batches that
are just large enough to produce three cylinders. Although care is taken to achieve
uniformity, we expect some variability among theb= 5 batches used to obtain the
following compressive strengths. (There is little reason to suspect interaction, and
hence only one observation is taken in each cell.) Data are also in the data set
sec95set2.rda.
Batch
Treatment B 1 B 2 B 3 B 4 B 5
A 1 52 47 44 51 42
A 2 60 55 49 52 43
A 3 56 48 45 44 38
(a)Use the 5% significance level and testHA:α 1 =α 2 =α 3 = 0 against all
alternatives.(b)Use the 5% significance level and testHB:β 1 =β 2 =β 3 =β 4 =β 5 =0
against all alternatives.9.5.11.Witha=3andb= 4, findμ, αi,βj andγij ifμij,fori=1, 2 ,3and
j=1, 2 , 3 ,4, are given by
67 712
10 3 11 8
85 9109.6 ARegressionProblem
There is often interest in the relationship between two variables, for example, a
student’s scholastic aptitude test score in mathematics and this same student’s