4. (10 points) Let V1, X2,.>, X20 be a random sample from Normal distribution with unknown parameters # and of that represent GRE scores of students from "Cosby Academy". Let Yi, Yz. .. I'd be a random sample from Normal distribution with unknown parameters /42 and ofrepresenting GRE scores of students from "Busby Academy". The samples are independent. and the variances are different. The following table shows the summary of samples scores: Summary Sample Mean Sample Variance Sample Size X & = 308.7 81 = 37 m = 20 4 = 306.5 $3 = 35 n = 10 (a) (4 points) Find the test statistic for testing Ho : /1 - /12 = 0 vs. H. : P1 - 12 > 0. What is the distribution of the test statistic under the null hypothesis [Note: If it is a t-distribution, provide its degrees of freedom] (b) (6 points) Using the above test statistic for Ho : #1 - (2 = 0 vs. Ha : HI - #2 > 0. find its p-value and conclude if you should reject the null hypotheses at level of significance, a = 0.05