Question 2 (20 pts) Consider the dataset discm,'eries containing the numbers great inventions and scientific discoveries in each year from 1860 to 1959. data ("discoveries 'v ) str (discoveries) ## Time-Series [1: 100] from 1860 to 1959: 5 3 0 2 0 3 2 3 6 1 ## creating a dataframe discoveries as. data. frame (cbind(1860: 1959 , discoveries)) colnames (discoveries) str (discoveries) ## 'data. frame' : $ year 100 num ## $ discoveries: num yearn , "discover ies obs . of 2 variables: 1860 1861 1862 1863 1864 5302032361 a) (2.5 pts) Plot the discoveries over time and comment on the trend, if any. b) (2.5 pts) Fit a Poisson response model with a constant term. Now compute the mean number of discoveries per year. What is the relationship between this mean and the coefficient seen in the model? c) (5 pts) Use the deviance from the model to check whether the model fits the data. What does this say about weather the rate of discovericns is constant over time? d) (5 pts) Fit a Poisson response model that is quadratic in the year. Test for the significance of the quadratic term. What does this say about the presence of a trend in discovery? e) (5 pts) Compute the predicted number of discoveries each year and show these predictions as a line drawn over the data. Comment on what you see.