EXAMPLE 7.15

Timing of food intake and weight loss. There is emerging evidence of a relationship between timing of feeding and weight regulation. In one study, researchers followed 402 obese or overweight individuals through a 20-week weight-loss treatment.²⁶ To investigate the timing of food intake, participants were grouped into early eaters and late eaters, based on the timing of their main meal. Here are the summary statistics of their weight loss over the 20 weeks, in kilograms (kg):

The early eaters lost more weight on average. Can we conclude that these two groups are not the same? Or is this observed difference merely what we could expect to see given the variation among participants?

While other evidence suggests that early eaters should lose more weight, the researchers did not specify a direction for the difference. Thus, the hypotheses are

H₀: μ₁ = μ₂

H_a: μ₁ ≠ μ₂

Because the samples are large, we can confidently use the t procedures even though we lack the detailed data and so cannot verify the Normality condition.

The two-sample t statistic is

= 3.71

The conservative approach finds the P-value by comparing 3.71 to critical values for the t(199) distribution because the smaller sample has 200 observations. Because Table D does not contain a row for 199 degrees of freedom, we will be even more conservative and use the first row in the table with degrees of freedom less than 199. This means we’ll use the t(100) distribution to compute the P-value.

Our calculated value of t is larger than the p = 0.0005 entry in the table. We must double the table tail area p because the alternative is two-sided, so we conclude that the P-value is less than 0.001. The data give conclusive evidence that early eaters lost more weight, on average, than late eaters (t = 3.71, df = 100, P < 0.001).