A university wants to compare out-of-state applicants' mean SAT math scores (μ₁) to in-state applicants' mean SAT math scores (μ₂). The university looks at 35 in-state applicants and 35 out-of-state applicants. The mean SAT math score for in-state applicants was 540, with a standard deviation of 20. The mean SAT math score for out-of-state applicants was 555, with a standard deviation of 25. It is reasonable to assume the corresponding population standard deviations are equal. At the 5% significance level, can the university conclude that the mean SAT math score for in-state students and out-of-state students differ? A) No, because the confidence interval contains zero. B) Yes, because the confidence interval contains zero. C) No, because the confidence interval does not contain zero. D) Yes, because the confidence interval does not contain zero.

Question

Quizplus · Accepted Answer

We will conduct a two-sample t-test with the following hypotheses: H0: μ₁ = μ₂ HA: μ₁ ≠ μ₂ The test statistic is: t = ((x1̄-x2̄)-(μ₁-μ₂))/S_pooled*sqrt(1/n1+1/n2) where x1̄ and x2̄ are the sample means, μ₁ and μ₂ are the population means, S_pooled is the pooled standard deviation, and n1 and n2 are the sample sizes. Using the given data, we have: x1̄ = 540 x2̄ = 555 S1 = 20 S2 = 25 n1 = n2 = 35 α = 0.05 degrees of freedom = 68 (df = n1 + n2 - 2) The pooled standard deviation is: S_pooled = sqrt(((n1-1)*S1^2 + (n2-1)*S2^2)/(n1+n2-2)) = sqrt(((34)*(20^2)+(34)*(25^2))/(68)) = 22.98 The calculated value of the test statistic is: t = ((540-555)-(0))/22.98*sqrt(2/35) = -2.61 The critical values for a two-tailed test with α = 0.05 and df = 68 are ±1.997. Since the calculated value of the test statistic (-2.61) falls outside the critical values (-1.997, 1.997), we reject the null hypothesis. Therefore, we can conclude that the mean SAT math score for in-state students and out-of-state students differ. The confidence interval does not contain zero.

A University Wants to Compare Out-Of-State Applicants' Mean SAT Math