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Do SAT coaching classes work? Do they help students improve their test scores?

Four students were randomly selected from those who completed an SAT coaching class. For each student, we recorded their first SAT score (before the class) and their second SAT score (after the class).

First SAT scores: 920, 830, 960, 910
Second SAT scores: 1010, 800, 1000, 980

If we do not know which scores belong to which students, which test should we use to analyze these data?

Answer :

To analyze whether SAT coaching classes help students improve their test scores, we need to compare the first and second set of SAT scores. The given data involves paired observations: each student's score before and after the coaching class.

Even though we do not know which first score corresponds to which second score, if we hypothetically could pair them, the appropriate statistical test would be the 'paired t-test.' This test is designed to compare the means of two related groups. In this case, the two groups would be each student's score before and after the SAT coaching class.

The steps to carry out a paired t-test would typically include:

  1. Establish hypotheses:

    • Null hypothesis (H0): There is no change in scores, i.e., the mean difference between the scores before and after the class is zero.
    • Alternative hypothesis (H1): There is a significant change in scores, i.e., the mean difference is not zero.

  2. Calculate the differences between paired observations.

  3. Compute the mean of these differences.

  4. Calculate the standard deviation of the differences.

  5. Compute the t-statistic, using the formula:
    [tex]t = \frac{\text{mean difference}}{\text{standard deviation of differences/sqrt(n)}}[/tex]
    where [tex]n[/tex] is the number of paired observations.

  6. Determine the critical t-value from t-distribution tables at the desired level of significance.

  7. Compare the computed t-statistic with the critical t-value to decide whether to reject or fail to reject the null hypothesis.

Because the pairing order of scores is unknown, a paired t-test cannot be performed directly with this dataset until the scores are matched for each specific student. However, typically, the test is insightful if students' scores can be matched, as it would allow an assessment of the individual improvements resulting from the coaching course.

To move forward, matching the scores per student would be the essential step if more information could be gathered to achieve this.

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