Identify the test statistic and P Value

Question

Researchers conducted an experiment to test the effects of alcohol. The errors were recorded in a test of visual and motor skills for a treatment group of 23 people who drank ethanol and another group of 22 people given a placebo. The errors for the treatment group have a standard deviation of 1.54, and the errors for the placebo group have a standard deviation of 1.36. Use a 0.05 significance level to test the claim that the treatment group has errors that vary more than the errors of the placebo groups Assume that both samples are independent simple random samples from populations having normal distributions. Find the test statistic F and P value

Answer 1

Hi Paula!

This is a test for equality of variances. It turns out that we can use an F statistic to determine the result. For formality, let X1 be the treatment group and X2 be the placebo group. What we want to test then are the following hypotheses:

H₀: σ₁² = σ₂²

H_A: σ₁² > σ₂²

We can test these hypotheses with the aforementioned F statistic:

F = s₁² / s₂²

which is distributed as an F statistic with df₁ degrees of freedom in the numerator and df₂ degrees of freedom in the denominator under the null hypothesis.

In this case, we get the result

F = (1.54²) / (1.36²) = 1.28

Because the sample sizes for the treatment and placebo groups are 23 and 22, the corresponding degrees of freedom are 22 and 21, respectively. Thus we have to look up a critical value for an F statistic corresponding to α = 0.05 with 22 and 21 degrees of freedom. This value is 2.07.

That is, if our obtained F statistic is greater than 2.07, we will reject the null hypothesis in favor of the alternative hypothesis, and if it is less than or equal to 2.07 we will fail to reject. Of course, F = 1.28 which is less than 2.07 so we would fail to reject the null hypothesis and conclude that there is no good reason to believe that the variance of the treatment group is greater than that of the placebo.