"In Exercises 13–18, test the claim about the difference betwe...

Question

"In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: σ₁² = σ₂²; α = 0.05.
Sample statistics: s₁² = 310, n₁ = 7 and s₂² = 297, n₂ = 8"

Claim: σ₁² = σ₂²; α = 0.05.

Sample statistics: s₁² = 310, n₁ = 7 and s₂² = 297, n₂ = 8"

Accepted Answer

Hello, everyone, let's take a look at this question together. A researcher wants to test whether the variance of test scores in School A is greater than that in School B at the alpha equals 0.05 significance level. The sample variances and sizes are given as sample variance 1 equals 520, sample size 1 equals 7 for school A, and sample variance 2 equals 410, and sample size 2 equals 8 for school B. Assume both populations are normally distributed and samples are independent. What is the correct decision regarding the claim that population variance 1 is greater than population variance 2? Is it answer choice A, We reject the null hypothesis. Answer choice B, we fail to reject the null hypothesis. Answer choice C, the test statistic is less than 1, or answer choice D, the critical value is less than the test statistic. So in order to solve this question, we have to recall how to test a claim so that we can test the claim that population variance 1 is greater than population variance 2 at the alpha equals 0.05 significance level. To determine what is the correct decision regarding that claim given our sample variances and sizes, and we can recall that the first step in testing our claim to determine the correct decision is to state the hypotheses, starting with our null hypothesis, which is population variance 1, is less than or equal to population variance 2. And then our alternative hypothesis is population variance 1 is greater than population variance 2, and then we can compute our test statistic using the formula for the test statistic F, which is equal to sample variance 1 divided by sample variance 2. And plugging in the values into the formula for the test statistic F, we have 520 divided by 410, giving us 1.2683 as the value for our test statistic. And then we can determine our degrees of freedom, starting with the degrees of freedom for the numerator, which is given as seven subtracted by 1, so our degrees of freedom 1 for the numerator is equal to 6. And then we can calculate the degrees of freedom for the denominator, which is equal to 8 subtracted by 1, giving us 7 as the value for our degrees of freedom 2, which is for the denominator. Therefore, we are looking for our critical value at alpha equals 0.05, for a right tail test with our degrees of freedom 1 equaling 6, and our degrees of freedom 2 equaling 7. And from our F tables, we can determine our critical value is approximately 3.97. So now we can compare our test statistic to our critical value, and since our test statistic of 1.2683 is less than our critical value of 3.97, we failed to reject the null hypothesis. So in conclusion, there is not enough evidence at alpha equals 0.05 to support the claim that population variance 1 is greater than population variance 2. And based on our decision regarding the claim, we can determine that answer choice B is the correct answer, as we fail to reject the null hypothesis, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

F-Test for Variances

Significance Level (α)

Watch next