Q. 1 Suppose that a we have a continuous response variable Y and a single categorical variable X with values 1,...,p. It is of interest to test if the levels of the response vary among categories of the predictor. Define the indicator variables Ej { b 1 if X = j for j=1,..., p otherwise and consider fitting the linear model Ya₁E+02E2++ apEp+E a) State the assumptions of this model and give an interpretation for the parameters. b) Use the matrix-based results for multiple regression to obtain the least squares estimates for 01,...,ap. c) Give the variance of â; and the correlation between â, and a when jk. d) Show that the usual unbiased estimator of o2 can be written as Σετ (1 - 1)3, n-p where n, is the number of subjects with a j and 83 is the sample variance of y for those subjects. e) Give a formula for the entries in the projection matrix P and have give the internally studen- tized residuals. f) State the null hypotheses that are being tested in the usual t-tests that are produced as part of the regression output and also the F test from the ANOVA for this model. Comment on whether these are appropriate tests.