Question

A university with 5000 undergraduates in each of three years selects 100 first-years and 80 final-years uniformly at random and administers a mathematics exam. It wishes to detect whether the average score in this mathematics exam among final-years is greater than the average score among first-years. (a) Choose a statistical test for this purpose. Give a full statement of the assumptions required for this test, and also the null and alternative hypotheses. 3 pts (b) Discuss the reasons why you think the assumptions are reasonable to make, and any assumptions you think are unnecessary or unreasonable. 2 pts (c) Give a formula for the relevant test statistic and its distribution under the null hypoth- esis. Also state the rejection region at the 0.5% significance level. (Note that this is much more stringent than the usual 5% significance level.) 3 pts

Answer 1

(a) Here we use 2 sample t test. We assume that two samples come from two independent normal populations. Moreover we assume that two population variances are equal. Let i true mean score in the mathematics exam for first-year students 12 true mean score in the mathematics eram for final – year students Null hypothesis, H:41 = H2 vs. Alternative hypothesis, He: M1 <H2. Hence it is left tailed test. (6) Since the math score is continuous variable and has symmetric belled shaped distribution. Hence it is reasonable to assume, two samples come from two independent normal populations. However assumption of equal variance is unreasonable.

T1 - T2 (c) Test statistic=t= Spymi + 1 n2 where, ni 100, n2 = 80, 71 sample mean score for first year students, 1 2 = sample mean score for final year students, si = sample sd for first year students, S2 = sample sd for final year students, sp = pooled sd = (nı – 1)s + (n2 – 1) s ni +n2 - 2 Under Ho, tot-distribution with degrees of freedom = n1 + n2 - 2 100+ 80 – 2 = 178. Critical value = t0.005,178 -2.6037 (U se R code: round(qt(0.005, 178), 4)). We reject Ho at 0.5% level of significance if t < Critical value.