$\backslash$table$[[,,,$Day$],[$Density$,$Moisture,Unit,$1,2,3],[1\mathrm{.}1,0\mathrm{.}10,2\mathrm{.}70,0\mathrm{.}34,0\mathrm{.}11],[,,2,2\mathrm{.}90,1\mathrm{.}57,1\mathrm{.}25],[,0\mathrm{.}20,3,5\mathrm{.}20,5\mathrm{.}04,3\mathrm{.}70],[,,4,3\mathrm{.}60,3\mathrm{.}92,2\mathrm{.}69],[,0\mathrm{.}24,5,4\mathrm{.}00,3\mathrm{.}47,3\mathrm{.}47],[,,6,4\mathrm{.}10,3\mathrm{.}47,2\mathrm{.}46],[1\mathrm{.}4,0\mathrm{.}10,1,2\mathrm{.}60,1\mathrm{.}12,0\mathrm{.}90],[,,2,2\mathrm{.}20,0\mathrm{.}78,0\mathrm{.}34],[,0\mathrm{.}20,3,4\mathrm{.}30,3\mathrm{.}36,3\mathrm{.}02],[,,4,3\mathrm{.}90,2\mathrm{.}91,2\mathrm{.}35],[,0\mathrm{.}24,5,1\mathrm{.}90,3\mathrm{.}02,2\mathrm{.}58],[,,6,3\mathrm{.}00,3\mathrm{.}81,2\mathrm{.}69],[1\mathrm{.}6,0\mathrm{.}10,1,2\mathrm{.}00,0\mathrm{.}67,0\mathrm{.}22],[,,2,3\mathrm{.}00,0\mathrm{.}78,0\mathrm{.}22],[,0\mathrm{.}20,3,3\mathrm{.}80,2\mathrm{.}80,2\mathrm{.}02],[,,4,2\mathrm{.}60,3\mathrm{.}14,2\mathrm{.}46],[,0\mathrm{.}24,5,1\mathrm{.}30,2\mathrm{.}69,2\mathrm{.}46],[,,6,0\mathrm{.}50,0\mathrm{.}34,0\mathrm{.}00]]($a$)$ Describe the study in terms of the between$-$subjects and within$-$subjects

designs.

$(5$ points$)$

$($b$)$ Compute the mean of the observations for each soil bulk density and soil

moisture level at each time of measurement, and make a profile plot of the

results for each treatment.

$(5$ points$)$

$($c$)$ Write a linear model for a split$-$plot analysis of variance, identify the terms, and

indicate the assumptions necessary for the analysis with this model.

$(5$ points$)$

$($d$)$ Conduct the split$-$plot analysis for the data, test the necessary hypotheses, and

compute treatment means and their standard errors. What are your conclusions?

$(5$ points$)$

$($e$)$ Obtain the residual plots from the split$-$plot analysis, and interpret them.

$(4$ points$)$

$($f$)$ Compute the sum of squares partitions for the linear and quadratic, contrasts on

time and their interactions with density and moisture treatments; test the null

hypotheses, and interpret the results.

$(6$ points$)$A soil scientist conducted an experiment to evaluate the effects of soil compaction and

soil moisture on the activity of soil microbes. Reduced levels of microbe activity will

occur in poorly aerated soils. The aeration levels can be restricted in highly saturated

or compacted soils. Treated soil samples were placed in airtight containers and

incubated under conditions conductive to microbial activity. The microbe activity in

each soil sample was measured as the percent increase in $CO2$ produced above

atmospheric levels.

The treatment design was a $33$ factorial with three levels of soil compaction $($bulk

density $=mg$ soil$/$cubic meter$)$ and three levels of soil moisture $(kg$ water $\frac{?}{kg}$ soil$).$

There were two replicate soil container units prepared for each treatment.

The CO$2$ evolution$/$kg soil$/$day was recorded on three successive days. The data for

each soil container unit are shown in the table.