4. A controlled experiment was conducted to study the effect of treatment A and the effect of treatment B on the expected value of the response variable y which is normally distributed. Treatment A has two levels, labeled a1, a2. Treatment B has three levels, labeled b1, b2, b3. The y-responses within and across the six cells (i.e., combination treatment levels) are statistically independent with constant variance whose value is unknown. Use any math/stat software (e.g., Page 4 www.numbergenerator.org/randomnumbergenerator) of your choice to find a random number generator to randomly select from the dataset Exam 2 Problem 4 data.txt" one row of each AB-combination level, e.g., A = a1, B = b1, etc. Place the six selected rows into the six cells, respectively, in Table 1, and N = 20 for each cell. Table 1 y: sample mean / B = b1 B = b2 B = b3 y: sample SD/ N A = al A = a2 SD standard deviation Perform analysis of variance to evaluate the effect of treatment A, the effect of treatment B, and possible interactions of the two treatments, by covering a) Write down a linear regression model corresponding to the analysis of variance in a), i.e., using indicator variables. [5 points] b) Construct the table of sums of squares for the regression analysis in d). [10 points] c) Will the table in b) change as defining the indicator variables changes? Provide mathematical proof of your answer. [10 points] d) Consider that treatment A and treatment B do not interact. Provide the best unbiased estimate of the difference in the conditional expectation of y between B = b3 and B = b1. [10 points] e) Consider that treatment A and treatment B interact. Provide the best unbiased estimate of the difference in the conditional expectation of y between B = b3 and B = b1. [10 points]