Toxicity experiment. In an experiment testing the effect of a toxic substance,

14.12. Toxicity experiment. In an experiment testing the effect of a toxic substance, 1,500 experimental insects were divided at random into six groups of 250 each. The insects in each group were exposed to a fixed dose of the toxic substance. A day later, each insect was observed. Death from exposure was scored 1, and survival was scored 0. The results are shown below; Xj denotes the dose level (on a logarithmic scale) administered to the insects in group j and Y.j denotes the number of insects that died out of the 250 (nj) in the group.j123456Xj:123456nj:250250250250250250Y.j:285393126172197Logistic regression model (14.20) is assumed to be appropriate.a. Plot the estimated proportions pj = Y.j/nj against Xj. Does the plot support the analyst’s belief that the logistic response function is appropriate?b. Find the maximum likelihood estimates of ?0 and ?1. State the fitted response function.c. Obtain a scatter plot of the data with the estimated proportions from part (a), and superimpose the fitted logistic response function from part (b). Does the fitted logistic response function appear to fit well?d. Obtain exp(b1) and interpret this number.e. What is the estimated probability that an insect dies when the dose level is X = 3.5?f. What is the estimated median lethal dose—that is, the dose for which 50 percent of the experimental insects are expected to die?Simple Logistic Regression ModelYi are independent Bernoulli random variables with expected values E{Yi} = ?i, where:(14.20)E{Y i }=? i (? 0 +? 1 X i )/(1+exp(? 0 +? 1 X i ))The X observations are assumed to be known constants. Alternatively, if the X observations are random, E{Yi} is viewed as a conditional mean, given the value of Xi.