Part B $$Two way ANOVA $$ Dental Health

The data set ToothGrowth has measurements of tooth growth $($len$)$ of guinea pigs for different dosages of Vitamin C $($dose$)$ and two different delivery methods $($supp$).$

To load the data set, ToothGrowth.txt$,$ change to the proper directory, and then execute the following command in R:

Tooth $<-$ read.table$("$ToothGrowth$.$txt$",$header$=$T$)$

attach$($Tooth$)$

Tooth contains three vectors: len, supp, and dose, as described in this table:

Column Descriptions of each column Name of each column in R

$1$ Tooth length len

$2$ Delivery method $($orange juice or ascorbic acid$)$ supp

$3$ Dosage level $(0\mathrm{.}5,1,$ or $2)$ in milligrams dose

Use the following commands to test whether dose and supp are treated as categorical variables. If the command responds $$FALSE$,$ use the factor$()$ command to make them categorical:

is$.$factor$($supp$)$

is$.$factor$($dose$)$

$1.$ Create a linear model for Tooth growth $($len$)$ in terms of the delivery method $($supp$),$ and fill in your answers below.

State the null hypothesis:

H$0$: The delivery method does not affect tooth growth in guinea pigs

Paste a copy of the anova output for this model here:

Paste a copy of the summary output for this model here:

$2.$ Create a linear model for Tooth growth $($len$)$ in terms of the dosage level $($dose$)$

State the null hypothesis:

H$0$: The dose of vitamin C does not affect tooth growth in guinea pigs

Paste a copy of the anova output for this model here:

Paste a copy of the summary output for this model here:

$3.$ Now create a linear model for Tooth growth $($len$)$ in terms of both the delivery method $($supp$)$ and the dosage level $($dose$)$ using the lm$()$ command:

model$<-$lm$($len~supp$+$dose$)$

State the overall null hypothesis:

H$0$:

Paste a copy of the anova output for this model here:

Paste a copy of the summary output for this model here:

$4.$ Use your outputs above to fill in the p$-$values and R$2$ values in the table below. $($Recall that the overall p$-$value can be found in the last line of the output from the summary$()$ command.$)$

Model p$-$value R$2$ value

len~supp

len~dose

len~supp$+$dose

$5.$ Compare the p$-$values and R$2$ values in the table above to determine which model best explains the variability in len. Report that model, here.

Model that best explains the variability in len: