l. The dataset discoveries in the faraway package lists the number of great inventions and scientific discoveries each year from 18601959. 3.. Create a. variable year =1860 :1959 and plot discoveries versus year . b. What is the correlation between discoveries and year '? c. Find the sample mean and sample variance ofdiscoveries and comment on whether it seems reasonable to view the dataset % 100 independent and identically distributed observations from a Poisson distribution. 2. Fit a Poisson regression model with discoveries as the dependent variable and yea as the independent variable. a. What is the pvalue for the Wald test of the null hypothesis tha = 0 versus a twoesided alternative hypothesis. b. Give an interpretation for g\" in terms of changes in expected discoveries each year. c. Find a 95 percent confidence interval forg . (1. Assuming that the tted model is correct and extrapolates to the future , nd the probability of observing 4 great discoveries in 1960. e. Does the deviance support that the model fits the data? Explain. 3.The esdcomp dataset in the faraway package provides data on the number of com plaints filed against emergency room physicians. It includes the total number of patient visits, a dummy variable for being a medical resident} a dummy variable for being male and the total number of hours worked. We will fit a rate model to examine factors affecting the rate of complaints Fit the model ratemod =g1m (complaints residency +gender +offset (log (visits ))+I(visits / hours J, family =poisson ,data =esdcomp } ' a. Provide an interpretaion for eacha in the model (not includinga'}. b. Which coefficients are significantly different from 0 at the 0.05 significance level? 4. Give the pvalue for the test of model fit based on the deviance statistic. 5.a. Find the sample mean and sample standard deviation of visits per hour. b. Accord ing to the estimated model , by what percentage does the rate of complaints change for every extra 0.25 visits per hour