Two suppliers, unequal delivery-time spreads: Welch's t-test

Question

A buyer compares delivery times (days). Supplier 1: 20 shipments, mean 5.0, sample standard deviation 1.0. Supplier 2: 15 shipments, mean 6.8, sample standard deviation 3.0. The spreads are very different, so use Welch's test.

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

Accepted Answer

Hypotheses: H0: \mu1 = \mu2 versus H1: \mu1 
eq \mu2. Standard error of the difference: 	ext{SE} = \sqrt{\frac{s1^2}{n1} + \frac{s2^2}{n2}} = \sqrt{\frac{1}{20} + \frac{9}{15}} = \sqrt{0.05 + 0.60} = \sqrt{0.65} = 0.8062. Test statistic: t = \frac{5.0 - 6.8}{0.8062} = \frac{-1.8}{0.8062} = -2.23. Welch degrees of freedom: 	ext{df} = \frac{0.65^2}{\dfrac{0.05^2}{19} + \dfrac{0.60^2}{14}} = \frac{0.4225}{0.0001316 + 0.0257143} \approx 16. The two-sided 5% critical value for about 16 degrees of freedom is roughly 2.120. Since -2.23 = 2.23 2.120, reject H0. The p-value is about 0.040. Conclusi…

Two suppliers, unequal delivery-time spreads: Welch's t-test

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