A study was carried out in $n=5$ small apple orchards. The $i-t$ th orchard has $r_i$ trees which were all given a dose of additive $x_i.$ After one month, $y_i,$ the number of trees that did not have certain fungus was recorded. The following is the data per orchard, presented in the format $(r_i,y_i,x_i)$:$(45,9,-1),(40,12,-0\mathrm{.}5),(45,21,0),(20,13,0\mathrm{.}5),(35,29,1).$ In this problem, the values of doses of additive are coded, hence the negative $x_i$ values.

The following are outputs from the analysis of these data after fitting a glm model with logit link and binomial distribution.

hat$({)}_0=-0\mathrm{.}0507,$hat$({)}_1=1\mathrm{.}4702$

The null deviance was $40\mathrm{.}8437$ on $4$ degrees of freedom.

The residual deviance was $0\mathrm{.}3805$ on $3$ degrees of freedom.

AlC: $23\mathrm{.}3102$

Examine the information given and do computations before answering the questions below.

A$)$ We use the null deviance to test

B$)$ Compute the p$-$value for comparing $($i$.$e$.$ testing$)$ the model with common probability $($null hypothesis$)$ against the maximal model $($alternative hypoyhesis$).$

C$)$ For the hypothesis described in B$),$ use the $p-$value you computed also in B$)$ and $=0\mathrm{.}05$ to select the correct statement below.

D$)$ Compute the log$-$likelihood of the maximal model. Hint: Use the R function $db$ inom and estimate the probability using the observed proportion of trees with no fungi in each orchard.

E$)$ Compute the predicted proportion of trees without fungus when the dose of additive is $x=1$ :