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Paired t-test: dietary product weight loss

Short answer

Paired t-test on weight losses (before − after): d̄ = 4.125 lb, t = 2.553. At α = 0.05, critical value t(0.05,7) = −1.895; since t > −1.895, we fail to reject H0. The data support the claim of average weight loss of at least 3 lb.

A liquid dietary product implies in its advertising that use of the product for one month results in an average weight loss of at least 3 pounds. Eight subjects use the product for one month, and the resulting weight loss data are reported below. Do the data support the claim of the producer of the dietary product with the probability of a type I error set to 0.05?

Subjects12345678
Weight before (lb)165201195198155143150187
Weight after (lb)161195192193150141146183

The same subjects are weighed before and after, so a paired t-test on the weight losses d = before − after is used. The claim (μ_d ≥ 3) is placed in H0, and rejection would mean the claim is not supported.

  1. H0: μ_d ≥ 3 lb (average loss is at least 3 lb, the claim) H1: μ_d < 3 lb (average loss is less than 3 lb) (left-tailed)
  2. Level of significance: α = 0.05
  3. Differences d (before − after):
d  = 4, 6, 3, 5, 5, 2, 4, 4
n  = 8, Σd = 33, d̄ = 4.1250
Σ(d−d̄)² = 10.8750, s_d² = 1.5536, s_d = 1.2464
t  = (d̄ − μ_d)/(s_d/√n) = (4.1250 − 3)/(1.2464/√8)
   = 1.1250/0.4407 = 2.553
  1. Critical value: t at α = 0.05, df = n−1 = 7: −1.895. Reject H0 if t < −1.895.
  2. Decision: the calculated value falls in the acceptance region (p-value = 0.9810), so fail to reject H0.

Conclusion: the data support the producer's claim that the average weight loss is at least 3 lb.

✓ Answer checked by Claude Sonnet 5.5

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