Regression Analysis — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

(iii) The correlation coefficient between X and Y. (iv) The mean values of X and Y. (v) The standard deviation of Y given standard deviation of X = 3. (vi) The angle between two lines of regression. 7. (a) For multiple linear regression model, define residual vector e. Show that Discuss the importance of R2 for model adequacy checking. (8,7) 8. (a) Suppose the postulated model is E(Y) = Bo + X. But the model E(Y) = Bo+ B,X + B,?is actually the true response function unknown to us. If we use observations of Y at X = -1, 0, 1, to estimate Bo and B, from the postulated model, what biases will be induced? (b) Find an unbiased estimate of error variance o' in multiple linear regression model. (8,7) (a) Define Karl Pearson's coefficient of correlation. If X and Y are random variables and a, b, c, d are any numbers such that a * 0, c7;0, then ac (b) Prove that for two independent variables, correlation coefficient r is 0. Is the converse true? Justify your answer with an example. Also find limits of r. (8,7) 2. (a) If X and Y are two variables with variances o} and o3, respectively and i is the coefficient of correlation between them. If U = X + KY and V = X+ 2 X. Find the value of lo so that t and V are uncorrelated. (b) The coefficient of rank correlation between the marks obtained by 10 students in Mathematics and Statistics was found to be 0.5. It was later discovered that the difference in ranks in two subjects obtained by one of the students was wrongly taken as 3 instead of 7. Find the correct coefficient of rank correlation. (8,7) 3. Consider the simple linear regression model y = Bo + B,x+8 where E(8) = 0, V(8) = o', e's are uncorrelated. (1) Show that E(MSR) = 02 + B} Sxx (ii) Show that E(MSE) = o' Hence test for significance of regression, where symbols have their usual meanings. (15) 4. For a given model Y, V(E) = 01, p(X) = P < n. Find the least squares estimate B of B. Show that (i) B is linear function of Y (it) B is unbiased for B 5. (a) Consider the model Obtain least square estimate of ß, + 2ß,. Also find its variance. (b) Find 100 (1-a) % confidence interval for o' in simple linear regression model. (9,6) 6. The equations of two lines of regression are Find the following: (i) The lines of regression of X on Y and Y on X. (i) Regression coefficients of Y on X and X on Y.