A study was carried out in n=6 small apple orchards. The ith orchard has ri trees which were all given a dose of additive xi . After one month, yi , the number of trees that did not have certain fungus was recorded. The following is the data per orchard, presented in the format (ri,yi,xi) : (30,22,-1), (40,27,-0.6), (45,23,-0.2), (20,10,0.2), (40,16,0.6), (25,12,1). In this problem, the values of doses of additive are coded, hence the negative xi values.

The following are outputs from the analysis of these data after fitting a glm model with logit link and binomial distribution. ^0=0.1761 , ^1=0.6602

The null deviance was 11.4689 on 5 degrees of freedom.

The residual deviance was 2.5381 on 4 degrees of freedom.

AIC: 29.7356

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

A) We use the residual deviance to test (select 1 of the following) -

1.the logistic regression model (null hypothesis) against the maximal model (Alternative hypothesis).

2. the model with common p (null) against maximal (alternative)

3. the model with common p (null) against logistic regression model(alternative)

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 hypothesis described in B), use the p-value you computed also in B) and =0.05 to select the correct statement below.

1. We reject the logistic regression model

2. We dont reject logistic regression model

D) Compute the log-likelihood of the maximal model. Hint: Use the R function dbinom 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 :