A guidance counselor claims that students who take the SAT twice will improve their scores the second time they take the SAT. The table shows both math SAT scores for 12 students who took the SAT twice. At \(\alpha=0.01\), can you support the guidance counselor's claim?

(a) identify the claim and state \(H_{0}\) and \(H_{a}\),

(b) find the critical value,

(c) find the test statistic,

(d) decide whether to reject or fail to reject the null hypothesis,

(e) interpret the decision in the context of the original claim.

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Student 1 2 3 4 5 6 Score on first SAT 440 510 420 450 620 450 Score on second SAT 440 570 510 470 610 450 Student 7 8 9 10 11 1 12 Score on first SAT 350 470 320 510 630 570 Score on second SAT 370 530 290 500 640 600