Two classes, unequal grade spreads: Welch's t-test

Question

Two classes take the same exam. Class A: 25 students, mean 78, sample standard deviation 8. Class B: 25 students, mean 72, sample standard deviation 16. The spreads differ sharply, so use Welch's test.

Run Welch's two-sample t-test at the 5% level (two-sided): do the mean scores differ? Report the t-statistic and its approximate degrees of freedom.

Accepted Answer

Hypotheses: H0: \muA = \muB versus H1: \muA 
eq \muB. Standard error of the difference: 	ext{SE} = \sqrt{\frac{sA^2}{nA} + \frac{sB^2}{nB}} = \sqrt{\frac{64}{25} + \frac{256}{25}} = \sqrt{2.56 + 10.24} = \sqrt{12.80} = 3.578. Test statistic: t = \frac{78 - 72}{3.578} = \frac{6}{3.578} = 1.68. Welch degrees of freedom: 	ext{df} = \frac{12.80^2}{\dfrac{2.56^2}{24} + \dfrac{10.24^2}{24}} = \frac{163.84}{0.2731 + 4.3691} \approx 35. The two-sided 5% critical value for about 35 degrees of freedom is roughly 2.030. Since 1.68 = 1.68 < 2.030, we fail to reject H0. The p-value is about 0.10. Concl…

Two classes, unequal grade spreads: Welch's t-test

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