2. 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 al, 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 o? whose value is unknown. Use any math/stat software (e.g., www.numbergenerator.org/randomnumbergenerator) of your choice to find a random number generator to randomly select from the dataset \"Exam 2 Problem 2 data.txt\" one row of each AB-comb\\ination level,e.g.,A=al,B= bl, etc. Place the six selected rows into the six cells, respectively, in Table 1, and N = 20 for each cell. Note: the raw data is not given; thus, it is not possible to use any statistical or mathematical software for statistical analysis. Show every step of your calculations for problems 3, 3d, and 3e. Table 1 V: sample mean / B=bl B=hb2 B=h3 y:sample SD / N A=al Y11./511 Y12./512 Y13./513 A=a2 Y21./521 Y22./S22 Y23./523 SD = standard deviation Instead of randomly sampling the data from the dataset, here is the representation of the dataset's corresponding value. Perform two-way analysis of variance and the corresponding linear regression analysis: a) Write down the two-way analysis of variance model and the corresponding linear regression model. Make sure that constraints are specified. [10 points] b ~ For the linear regression analysis with the model in a), will changing the indicator variables change the table of sums of squares? Provide mathematical proof of your answer. [10 points] c) Suppose that treatment A and treatment B are known not to interact with each other. Provide the best unbiased estimate and 95% confidence interval for the difference in the conditional expectation of vy between B = bl and Bave = (b2+b3)/2 at each level of treatment A with mathematical proof. [15 points] d) In contrast to c), suppose that treatment A and treatment B are known to interact with each other. Provide the best unbiased estimate and 95% confidence interval for the difference in the conditional expectation of y between B = b1 and Bave = (b2+b3)/2 at each level of treatment A with mathematical proof. [15 points]