Question

8. An SAT prep course claims to improve the test score of students. The table below shows the scores for seven students the first two times they took the verbal SAT. Before taking the SAT for the second time, each student took a course to try to improve his or her verbal SAT scores. Do these results support the claim that the SAT prep course improves the students' verbal SAT scores?

Let d=(verbal SAT scores prior to taking the prep course)−(verbal SAT scores after taking the prep course)d=(verbal SAT scores prior to taking the prep course)−(verbal SAT scores after taking the prep course). Use a significance level of α=0.01 for the test. Assume that the verbal SAT scores are normally distributed for the population of students both before and after taking the SAT prep course.

Student	1	2	3	4	5	6	7
Score on first SAT	430	500	420	380	560	450	510
Score on second SAT	480	570	480	420	580	530	530

Step 1 of 5: State the null and alternative hypotheses for the test.

Ho: μd (=,≠,<,>,≤,≥) 0

Ha: μd (=,≠,<,>,≤,≥) 0

Step 2 of 5: Find the value of the standard deviation of the paired differences. Round your answer to one decimal place.

Step 3 of 5: Compute the value of the test statistic. Round your answer to three decimal places.

Step 4 of 5: Determine the decision rule for rejecting the null hypothesis H0H0. Round the numerical portion of your answer to three decimal places.

Reject Ho if (t, I t I), (<,>) _____

Step 5 of 5: Make the decision for the hypothesis test.

Reject Null Hypothesis Fail to Reject Null Hypothesis

Answer 1

Let us denote the difference

d = Score on second SAT - Score on first SAT

To test against

Here

sample mean of difference

sample standard deviation of difference

and sample size

The test statistic can be written as

which under H₀ follows a t distribution with n-1 df.

We reject H₀ at 5% level of significance if

Now,

The value of the test statistic

and critical value

Since , we reject H₀ at 5% level of significance and we can conclude that there is enough evidence to support the claim that the SAT prep course improves the students' verbal SAT scores.