Kindly address all the questions. Thanks

The table below gives the frequency of a critical illness disease by age group in a certain study. The table also gives the age midpoint (x), the number of people in each group (n), and y = lug _ , where 0 denotes the proportion in an age group with the disease. Contracted disease? Age group Yes No 20-25 25 9 10 -2.19722 30-34 32.5 2 13 5 -1.87180 35-39 37.5 3 9 2 -1.09861 40-44 42.5 5 10 15 -0.69315 45-49 47.5 6 7 13 -0.15415 50-54 52.5 5 3 0.51083 55-59 57.5 13 4 17 1.17865 60 69 65 10 1.38629 [x - 360; [x - 17437.5; [v - -2.9392; [v--13.615; Exy- -9.0429 (1) Calculate an estimate of the probability of having the disease under the assumption that the probability is the same for all age groups. [1] Consider the hypothesis that there are no differences in the probability of having the disease for the different age groups. (ii] (a) Construct an 8 x 2 contingency table which includes the expected frequencies under this hypothesis. (b) Conduct a y' test to investigate the hypothesis. [6] Consider the linear regression model y = a + Br + , where the error terms (:) are independent and identically distributed following a N(0, s') distribution. (iii) (a) Draw a scatterplot of y against x and comment on the appropriateness of the considered model. (b) Calculate the fitted regression line of y on r. (9) Calculate a 99% confidence interval for the slope parameter. (d) Interpret the result obtained in part (ii) with reference to the confidence interval obtained in part (iii) (c). [14] [Total 21]An actuary is considering statistical models for the observed number of claims, X. which occur in a year on a certain class of non-life policies. The actuary only considers policies on which claims do actually arise. Among the considered models is a model for which P( X = x)=- 1=1, 2, 3, ... log[1-0) x where 0 is a parameter such that 0