question in pictures below:

6. A Two-Sample Test on Standardized Te st Scores N Bookmark this page {a} 3 points possible [graded] The National Assessment of Educational Progress tested a simple random sample of 1000 thirteen year old students in both 2M4 and 2008 and recorded each student's score. The average and standard deviation in 2004 were 25? and 39. respectively. In 2008. the average and standard deviation were 260 and 38.. respectively. Your goal as a statistician is to assess whether or not there were statistically signicant changa in the average test scores of students from 2004 to 2008. To do so. you make the following modeling assumptions regarding the test scores: - X1, ..., leepresent the scores in 2004. - X1, , leare iid Gaussians with standard deviation 3'). - [E [X1] = m,which is an unknown parameter. - Y1, ,Ymoaepresent the scora in 2008. - Y1, , Yloomre iid Gaussians with standard deviation 3%. - E [Y1] = #2. which is an unknown parameter. - X1,.._,Xnareindependenton1,.._,Y,r You dene your hypothesis test in terms of the null Ho 2 #1 = #2 {signifying that there were not signicant manges in test scores} and H1 : ,ul # #2. You design the test 2 m) . (.;.. where an represents the 1 qquantile of a standard Gaussian. Yr?\" V382+3 Hint: Under Hg : F1 = #3. the test statistic is distributed as a standard Gaussian: 317 V 382 + 392 a ~ Non You are encouraged to check this.(Compute the mean and variance and recall that the sum of iid Gaussians is again Gaussian'] What is the largest possible value of :7 so that at has level 10%? :3: If W is designed to have level 10%, would you reject or fail to reject the null hypothesis given the 2008 data? 0 Reject Q Fail to reject What is the p-value for this data set