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

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 = 245, n₁ = 31 and s₂² = 112, n₂ = 28

Claim: σ₁² ≠ σ₂²; α = 0.05.

Sample statistics: s₁² = 245, n₁ = 31 and s₂² = 112, n₂ = 28

Accepted Answer

Hello there. Today we're going to solve the following practice from together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given two independent random samples from normal populations, test at alpha equals 0.05. The claim that sigma 1 squad is less than or equal to sigma 22. The sample statistics are S12 is equal to 670, N1 is equal to 15, and S22 is equal to 690, N2 is equal to 12. What is the outcome? Awesome. So it appears for this particular prong, we're asked to take all the information that is provided to us by the prom itself, and we're asked to test the claim that sigma 1 squared is less than or equal to sigma 22. And we're asked to determine what is the outcome. So we're looking at our multiple choice answers, and we're asked to determine which of our multiple choice answers reflects the correct outcome. So with that in mind, let us read off our multiple choice answers to see what our final answer might be. A is do not reject H0. There is evidence that sigma 1 squad is less than sigma 22. B is reject H0, not enough evidence that sigma 1 squad is greater than sigma 22. C is reject H0. There is evidence that sigma 1 squad is greater than sigma 22. And finally, D is do not reject H0. Not enough evidence that sigma 1 squad is greater than sigma 22. Awesome. So firstly, in order to solve this problem, we need to pay attention to what our hypotheses are for this particular problem, starting with the null hypothesis, which we should note and recall that H0 denotes the null hypothesis, and the null hypothesis for this particular problem states that sigma 12 is less than or equal to sigma 22. And the alternative hypothesis, which will denote as HA, the alternative hypothesis for this particular problem states that sigma 1 squared is greater than sigma 22. Fantastic. So now our second step is we need to determine the test statistic, which, as we should recall, F is equal to S1 2 divided by S22. Which is equal to 670 divided by 690, which will mean that F is equal to 0.9710 when we round to four decimal places. Our third step now is we need to determine the degrees of freedom for both of our samples. So DF1, the degrees of freedom for our first sample is equal to 14, and the degrees of freedom for our second sample, DF2 is equal to 11. Fantastic. So DF1 is 14 and DF2 is 11. So now, Our fourth step is we need to determine the critical value at the significance level alpha equals 0.05, which will mean that our critical value F subscript 0.0514.11 will be approximately equal to 2.74. Our fifth step now is we need to make our comparison, so we need to take 0.9710, and we need to note that it is less than 2.74. And because of this fact, because 2.74 is greater than 0.9710, we can state for our final conclusion that we do not reject the Noel hypothesis. There is not enough evidence that sigma 1 squared is greater than sigma 2 squared. Which will mean that our final answer without a doubt will have to be multiple choice answer letter D, which once again is do not reject H0, the null hypothesis, so do not reject the null hypothesis, not enough evidence that sigma 1 squared is greater than sigma 2 squared. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Hypothesis Testing

F-Test for Variances

Significance Level (α)

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