Two machines, unequal weight spreads: Welch's t-test

Question

A plant compares two filling machines (weights in grams). Machine A: 50 units, mean 200, sample standard deviation 5. Machine B: 30 units, mean 203, sample standard deviation 12. The spreads differ, so use Welch's test.

Run Welch's two-sample t-test at the 5% level (two-sided): do the mean weights 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{25}{50} + \frac{144}{30}} = \sqrt{0.50 + 4.80} = \sqrt{5.30} = 2.302. Test statistic: t = \frac{200 - 203}{2.302} = \frac{-3}{2.302} = -1.30. Welch degrees of freedom: 	ext{df} = \frac{5.30^2}{\dfrac{0.50^2}{49} + \dfrac{4.80^2}{29}} = \frac{28.09}{0.005102 + 0.794483} \approx 35. The two-sided 5% critical value for about 35 degrees of freedom is roughly 2.030. Since -1.30 = 1.30 < 2.030, we fail to reject H0. The p-value is about 0.20. C…

Two machines, unequal weight spreads: Welch's t-test

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