(a) 3 points possible (graded) The National Assessment of Educational Progress tested a simple random sample of 1000 thirteen year old students in both 2004 and 2008 and recorded each student's score. The average and standard deviation in 2004 were 257 and 39, respectively. In 2008, the average and standard deviation were 260 and 38, respectively. four goal as a statistician is to assess whether or not there were statistically significant changes in the average test scores of students from 2004 to 2008. To do so, you make the following modeling assumptions regarding the test scores: X1yes A joon represent the scores in 2004. X1,. . 1000 are iid Gaussians with standard deviation 39. . E X1] = #1. which is an unknown parameter. . Yl,. ..; Yloop represent the scores in 2008 Y1,...; Y10on are iid Gaussians with standard deviation 38. . E [Yi] = /2 which is an unknown parameter. . X1,. . . ) Xn are independent of Yl, . .., Yn- You define your hypothesis test in terms of the null Ho : /1 = /2 (signifying that there were not significant changes in test scores) and H1 : /1 / /2. You design the test 1 = 1 Vn/ Xn - Y. |2 V382 4 39 2 In/2) where qn represents the 1 - 7/ quantile of a standard Gaussian. Hint: Under Ho : /1 = /2 the test statistic is distributed as a standard Gaussian: Vn An Ym ON N (0, 1) V 382 + 393 You are encouraged to check this.(Compute the mean and variance and recall that the sum of lid Gaussians is again Gaussian.) What is the largest possible value of 7 so that i has level 10%? 7 = If w is designed to have level 10%, would you reject or fail to reject the null hypothesis given these data? Reject O Fail to reject What is the p-value for this data set