1. A toxicologist was interested in the effects of an insecticideon the mortality of a certain type of beetle. Six groups of beetleswere exposed to various concentrations (C) of the insecticide. Atthe end of the experiment dead beetles in each group werecounted.

concentration	10.8	11.6	12.1	12.6	13.1	13.5
death	15	24	26	24	29	29
group size	50	49	50	50	50	49

Consider the logistic regression model

logit [P(death at concentration C)] = β0 + β1 ×C

1. Write down the contribution to the likelihood function from thefirst group (that is, the group with death=15, size=50 and C =10.8).
2. Re-write the above for the log-likelihood function.
3. Write down the contribution to the likelihood function from thelast group (that is, the group with death=29, size=49 and C =13.5).
4. Re-write the above for the log-likelihood function.
5. Use proc fcmp to write a SAS program that computes thelog-likelihood function for the above application. In other words,your function accepts two numbers as values of β0 and β1, andoutputs the log-likelihood function evaluated at the prescribedinput values. Use the testing values β0 = 2 and β1 = 3 to make surethat you function works properly (it should give you a value of−5795.3).
6. Use SAS proc logistic to find the ML estimates and thecorresponding maximized log-likelihood function value. Use your ownfunction developed in part (5) to verify the SAS results.
7. By adding or subtracting 10% from the ML estimates, you cancreate four pairs of (β0,β1). Compute the log-likelihood functionfor all these 4 pairs of parameter values. Can you find a largerlog-likelihood function value than that evaluated at the MLestimates? Why?

2) Revisit the toxicology example: six groups of beetles wereexposed to various concentrations

(C) of the insecticide. At the end of the experiment deadbeetles in each group were counted.

concentration	10.8	11.6	12.1	12.6	13.1	13.5
death	15	24	26	24	29	29
group size	50	49	50	50	50	49

Consider the logistic regression model

logit[P(death at concentration C)] = β0 + β1 X C

1. Under the assumption β1 = 0, write down the log-likelihoodfunction (of _0 alone).
2. Write a SAS program to draw the above log-likelihoodfunction.
3. Use SAS to find the ML estimate β0 under the assumption β1 =0.
4. Use your own SAS program to find the maximized log-likelihoodfunction value at the above ML estimate of _0. Compare your resultswith what SAS reports.
5. Using results from Homework 2, calculate the likelihood ratiotest statistics for testing H0 : β1 = 0 vs. HA : β1 ≠0.
6. What is the p-value of the above?
7. Calculate the OR comparing two concentrations, one of which is0:4 units higher than the other.
8. Calculate the estimated probability of beetle death atconcentration C = 12:4.
9. Use SAS to draw a graph relating probability of beetle death toconcentration.