A tutor promises to improve GMAT scores of students by more than 50 points after three lessons. To see if this is true, the tutor takes a sample of 49 students' test scores after and before they received tutoring. The mean difference was 53 points better after tutoring, with a standard deviation of the difference equal to 12 points. Let µ_D denote the mean of the difference: score after tutoring minus score before tutoring. Which of the following is the appropriate conclusion at the 5% significance level? A) Reject H₀; we can conclude that the mean difference in test scores after and before tutoring is greater than 50 points. B) Reject H₀; we cannot conclude that the mean difference in test scores after and before tutoring is greater than 50 points. C) Fail to reject H₀, we can conclude that the mean difference in test scores after and before tutoring is greater than 50 points. D) Fail to reject H₀; we cannot conclude that the mean difference in test scores after and before tutoring is greater than 50 points.

Question

Quizplus · Accepted Answer

We need to conduct a one-sample t-test to determine whether the mean difference in test scores after and before tutoring is significantly greater than 50 points. The null hypothesis is that the mean difference is less than or equal to 50 points (H₀: µ_D ≤ 50). The alternative hypothesis is that the mean difference is greater than 50 points (HA: µ_D > 50). We can use the formula t = (x̄ - μ) / (s / √n) to calculate the test statistic, where x̄ is the sample mean difference (53), μ is the hypothesized population mean difference under the null hypothesis (50), s is the sample standard deviation (12), and n is the sample size (49). t = (53 - 50) / (12 / √49) = 4.04 Using a t-distribution table with 48 degrees of freedom (df = n - 1), we can find the critical value for a one-tailed test at the 5% significance level. The critical value is 1.677. Since our calculated t-value (4.04) is greater than the critical value (1.677), we reject the null hypothesis and conclude that the mean difference in test scores after and before tutoring is significantly greater than 50 points. Therefore, Choice A is the best answer.

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