(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.
Your 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:
represent the scores in 2004.
are iid Gaussians with standard deviation .
, which is an unknown parameter.
represent the scores in 2008.
are iid Gaussians with standard deviation .
, which is an unknown parameter.
are independent of .
You define your hypothesis test in terms of the null (signifying that there were not significant changes in test scores) and . You design the test
where represents the quantile of a standard Gaussian.
Hint: Under , the test statistic is distributed as a standard Gaussian:
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 so that has level ?
unanswered
If is designed to have level , would you reject or fail to reject the null hypothesis given these data?
Reject
Fail to reject
unanswered
What is the p-value for this data set?
unanswered