Discussion
The following are percentages of fat found in 5 samples of each of two brands of food :
A : 5.7, 4.5, 6.2, 6.3, 7.3
B : 6.3, 5.7, 5.9, 6.4, 5.1
To test the hypothesis of equal average fat content in the two types of food. Which of the following procedures is appropriate ?
Answer: A
The appropriate test is a two sample t-test with 8 degrees of freedom.
The two sets of five samples come from two different brands of food. There is no natural pairing between the first sample of brand A and the first of brand B, so the samples are independent, and a paired t-test is not appropriate. With small samples and unknown variances, the means are compared by the two sample (pooled) t-test, whose degrees of freedom are:
\[ \begin{aligned} df &= n_{1} + n_{2} - 2 \\ &= 5 + 5 - 2 = 8 \end{aligned} \]
9 degrees of freedom subtracts only one. The paired options would be right only if each A value were matched with a B value, as in before-and-after measurements on the same units; a paired test on 5 pairs would have 4 degrees of freedom.
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