Vitamin A Supplements in Low-Birth-Weight Babies 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, 130 low-birth-weight babies were randomly divided into two groups. Subjects in group 1 (the treatment group, n1=65) were given 25,000 IU of vitamin A on study days 1, 4, and 8 where study day 1 was between 36 and 60 hours after delivery. Subjects in group 2 (the control group, n2=65) 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 17.07 μg/dL. The control group had a mean serum retinol concentration of 12.88 μg/dL, with a standard deviation of 6.48 μg/dL. Does the treatment group have a higher standard deviation for serum retinol concentration than the control group at the α=0.01 level of significance? It is known that serum retinol concentration is normally distributed. 

Verified step by step guidance

1. Identify the hypotheses for the test comparing two population variances. Since we want to know if the treatment group has a higher standard deviation than the control group, the hypotheses are: \(H_0: \sigma_1^2 \leq \sigma_2^2\) (treatment variance less than or equal to control variance) \(H_a: \sigma_1^2 > \sigma_2^2\) (treatment variance greater than control variance).
2. Determine the test statistic to use. Because the data are normally distributed and we are comparing variances, use the F-test for equality of variances. The test statistic is \(F = \frac{s_1^2}{s_2^2}\) where \(s_1^2\) and \(s_2^2\) are the sample variances of the treatment and control groups respectively.
3. Calculate the sample variances by squaring the given standard deviations: \(s_1^2 = (17.07)^2\) \(s_2^2 = (6.48)^2\).
4. Compute the F-statistic by dividing the treatment group variance by the control group variance: \(F = \frac{s_1^2}{s_2^2}\).
5. Determine the critical value from the F-distribution table at the significance level \(\alpha = 0.01\) with degrees of freedom \(df_1 = n_1 - 1 = 64\) and \(df_2 = n_2 - 1 = 64\). Compare the calculated F-statistic to this critical value to decide whether to reject the null hypothesis.

Key Concepts

Hypothesis Testing for Variances

This involves comparing the variability of two populations to determine if their variances differ significantly. The null hypothesis typically states that the variances are equal, while the alternative suggests one variance is greater. Tests like the F-test are used to assess this by comparing sample variances under a chosen significance level.

F-Test for Equality of Variances

The F-test compares the ratio of two sample variances to determine if they come from populations with equal variances. It assumes normality of data and independent samples. The test statistic follows an F-distribution with degrees of freedom based on sample sizes, and the result is compared to critical values at a given significance level.

Significance Level and Decision Rule

The significance level (α) defines the threshold for rejecting the null hypothesis, commonly set at 0.01 or 0.05. If the test statistic falls into the critical region beyond this threshold, the null hypothesis is rejected. This controls the probability of a Type I error, ensuring conclusions are statistically reliable.