Hypothesis Tests for Comparing Variances
On the basis of data reported in a 1979 article in the Journal of Gerontology (“Serum
Ferritin in Elderly Population,” p. 521 – 524), the authors concluded that the ferritin distribution in
the elderly had a smaller variance than in the younger adults. (Serum ferritin is used in diagnosing
iron deficiency.) For a sample of 28 elderly men, the sample standard deviation of serum ferritin
(mg/L) was 𝑠1 = 52.6; for a sample of 26 young men, the sample standard deviation was 𝑠2 = 84.2.
Do the data support the conclusion applied to men? Let 𝜎𝐸2 denote the variance of the serum ferritin
distribution for the population of elderly men, and let 𝜎𝑌2 denote the variance of the serum ferritin
distribution for the population of young men. The authors’ claim is that 𝜎𝐸2 < 𝜎𝑌2 . We want to test
the claim at the 0.01 level of significance.
Step 1: H0: 𝜎𝐸2 ≥ 𝜎𝑌2 .
versus
Ha: 𝜎𝐸2 < 𝜎𝑌2 .
Step 2: 𝑛𝐸 = 28, 𝑛𝑌 = 26,  = 0.01
𝑆2
Step 3: The test statistic is 𝐹 = 𝑆𝐸2 , which under H0 has an F distribution with numerator d.f. = 27
𝑌
and denominator d.f. = 25.
Step 4: The rejection region is a left-hand tail of the F-distribution. The area of this left-hand tail is
0.01, our chosen significance level. The boundary point of the rejection region is 𝐹27,25,0.99 =
0.394.
Step 5: Now, we select the random sample from the population, collect the data, and do the
52.62
calculations. We find that 𝑓 = 84.22 = 0.390254.
Step 6: We reject H0 at the 0.01 level of significance. We have sufficient evidence to conclude that
the variance of serum ferritin in elderly men is lower than the variance of serum ferritin in young
men.