Question

Low-birth-weight babies are at increased risk of respiratory infections in the first few months of life and have low liver stores of vitamin A. In a randomized, double-blind experiment, 55 low-birth-weight babies were randomly divided into two groups. Subjects in group 1 (treatment group, n1 = 30) were given 25,000 IU of vitamin A on study days 1, 4, and 8; study day 1 was between 36 and 60 hours after delivery. Subjects in group 2 (control group, n2 = 25) were given a placebo. The treatment group had a mean serum retinol concentration of 45.77 micrograms per deciliter (μg/dL), with a standard deviation of 13.42 μg/dL. The control group had a mean serum retinol concentration of 15.88 μg/dL, with a standard deviation of 7.59 μg/dL. It is known that serum retinol concentrations are normally distributed. Determine if there is a difference in the standard deviation of serum retinol concentrations between the treatment group and the control group at the α = 0.05 level of significance?

Let σ1 denote the standard deviation of serum retinol concentrations for infants receiving vitamin A supplements, and σ2 the standard deviation for infants receiving a placebo.

Rejection Region:
We are performing a two-tailed test.
The critical value in the right tail is F_α/2 = .
The critical value in the left tail is F_1-α/2 = .
(Report the right-tail value as it appears in the table. Report the left-tail value rounded to 3 decimal places.)

Test Statistic:
The test statistic for this test is F₀ = .

Conclusion:
We  (reject / fail to reject) H₀.
The data  (does / does not) provide significant evidence of a difference in the standard deviations of serum retinol concentrations between infants who receive vitamin A supplements and those who receive a placebo.

P-Value:
Using Minitab Express, the P-value for this test is .

Refer to the Vitamin A Supplements data given in Question 3. Construct a 95% confidence interval for the ratio σ₁/σ₂ of standard deviations of serum retinol concentrations for infants who received vitamin A supplements and those who received a placebo. Report your answers rounded to 3 decimal places.

We are 95% confident that the standard deviation of serum retinol concentrations for infants who receive vitamin A supplements is between  (lower) and  (upper) times the standard deviation for infants who receive a placebo.

Answer 1

Sample 1:

s₁ = 13.42, n₁ = 30

Sample 2:

s₂ = 7.59, n₂ = 25

α = 0.05

Null and alternative hypothesis:

Hₒ : σ₁ = σ₂ ; H₁ : σ₁ ≠ σ₂

Degree of freedom:

df₁ = n₁-1 = 29

df₂ = n₂-1 = 24

Critical value(s):

Lower tailed critical value, Fα/₂ = F.INV(0.05/2, 29, 24) = 0.4643

Upper tailed critical value, F₁-α/₂ = F.INV((1-0.05)/2, 29, 24) = 2.2174

Test statistic:

F = s₁² / s₂² = 13.42² / 7.59² = 3.1262

Conclusion:

Reject the null hypothesis.

The data does provide significant evidence of a difference in the standard deviations of serum retinol concentrations between infants who receive vitamin A supplements and those who receive a placebo.

P-value = 2*F.DIST.RT(3.1262, 29, 24) = 0.0057

95% Confidence interval:

Lower Bound = (s₁² / s₂²)/F₁-α/₂ = 1.410

Upper Bound = (s₁² / s₂²)/Fα/₂ = 6.734