# (4) [14 pts] Grades on a standardized test are known to have a mean of 1000...

###### Question:

(4) [14 pts] Grades on a standardized test are known to have a mean of 1000 for students in Canada. The test is administered to 453 randomly selected students in Alberta; in this sample, the mean is 1013 and the sample standard deviation is 108.
(c) Another 503 students are selected at random from Alberta. They are given a 3-hour preparation course before the test is administered. Their average test score is 1019 with a standard deviation of 95. (i) Construct a 95% confidence interval for the change in average test scores as- sociated with the prep course. NOTE: You may recall from your statistics course learning about a test for the difference of means across two samples. In this case, our es- timator is not the sample means themselves, but the DIFFERENCE between the sample means, call it A -X1 - X2. Moreover, in order to perform our test of the hypothesis that the difference between the means is zero, we need to know what is the standard deviation of the difference between the sample means, or the square root of Var(A) Similarly as we saw in the previous question: Var (A) - Var(X1 - X2) - Var(X2) + Var(X2) since the two samples are independent of each other. (ii) Is there statistically significant evidence that the prep course helped?
(d) The original 453 students are given the prep course and then are asked to take the test a second time. The average change in their test scores is 9 points, and the standard deviation of the change is 60 points. (i) Construct a 95% confidence interval for the change in average test scores. (ii) Is there statistically significant evidence that students will perform better on their second attempt after taking the prep course? i) Students may have performed better in their second attempt because of the prep course or because they gained test-taking experience in their first attempt. Describe an experiment that would quantify these two effects.

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