A study was carried out in $n=6$ small apple orchards. The $i-t$ h 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 $(r_i,y_i,x_i)$:$(20,14,-1),(25,13,-0\mathrm{.}6),(45,28,-0\mathrm{.}2),(35,19,0\mathrm{.}2),(20,4,0\mathrm{.}6),(20,6,1).$ 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$({)}_0=0\mathrm{.}0102,$hat$({)}_1=-0\mathrm{.}8979$

The null deviance was $17\mathrm{.}2517$ on $5$ degrees of freedom.

The residual deviance was $6\mathrm{.}1712$ on $4$ degrees of freedom.

AlC: $31\mathrm{.}7361$

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

A$)$ We use the residual deviance to test

B$)$ Compute the p$-$value for comparing $($i$.$e$.$ testing$)$ the logistic regression model $($null hypothesis$)$ against the maximal model $($alternative hypothesis$).$

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

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We do not reject the logistic regression model

We reject the logistic regression model lint: 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 $x=-0\mathrm{.}2$ :