Question

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	470	470	510	410	400	390	530
Score on second SAT	500	490	580	490	420	430	590

1. State the null and alternative hypotheses for the test.

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

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

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

5. Make the decision for the hypothesis test.

Answer 1

(1) The Hypothesis

H0: = 0

Ha: > 0

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2) For The Standard deviation of the differences

Calculation for the mean and standard deviation:

Mean = Sum of observation / Total Observations

Standard deviation = SQRT(Variance)

Variance = Sum Of Squares (SS) / n - 1, where SS = SUM(X - Mean)².

#	Difference	Mean	(X-Mean)2
1	30	45.71	246.8041
2	20	45.71	661.0041
3	70	45.71	590.0041
4	80	45.71	1175.8041
5	20	45.71	661.0041
6	40	45.71	32.6041
7	60	45.71	204.2041

n	7
Sum	320
Mean	45.71
SS	3571.4287
Variance	595.2381
Std Dev	24.4

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(3) The Test Statistic:

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(4) The critical value at = 0.01, df = n - 1 = 6 is 3.143

The decision rule is Reject H0, if t obs > 3.143

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(5) Since t observed is > 3.143, reject H0.

there is sufficient evidence at the = 0.01 to conclude that SAT prep course improves the students verbal SAT scores.

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