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 = 0.01.
Sample statistics: s₁² = 842, n₁ = 11 and s₂² = 836, n₂ = 10

Claim: σ₁² ≤ σ₂²; α = 0.01.

Sample statistics: s₁² = 842, n₁ = 11 and s₂² = 836, n₂ = 10

Accepted Answer

Hello there. Today we're gonna solve the following practice problem 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. At the alpha equals 0.025 significance level, test the claim that sigma 1 squad is less than or equal to sigma 2 squad for two normal populations, sample data. S1 squared is equal to 950, N1 is equal to 9, S22 is equal to 890, N2 is equal to 8. What is the correct conclusion? Awesome. So it appears for this particular problem, we're asked to look at our multiple choice answers, and we're asked to determine which of our multiple choice answers presents the correct conclusion based on all the information that is provided to us by the prom itself. So now that we know what we're ultimately trying to solve or we're trying to figure out what the correct conclusion is, let's take a moment to read off our multiple choice answers to see what our final answer might be. A is reject H0. There is evidence that sigma 1 squad is greater than sigma 2 squad. B is do not reject H0. Not enough evidence that sigma 12 is greater than sigma 22. C is reject H0. Not enough evidence that sigma 1 squad is greater than sigma 22. And finally, D is do not reject H0. There is evidence that sigma 1 squared is less than sigma 22. Awesome. And let us quickly recall that H0 denote that this value H0 denotes the null hypothesis. So with that in mind, our first step in order to solve this particular problem is we need to pay attention to our hypotheses, starting with the null hypothesis for this particular prompt, and let us know that the null hypothesis for this particular problem will be that sigma 1 squared. is less than or equal to sigma 22, and that our alternative hypothesis HA will state that sigma 1 squad is greater than sigma 2 squad. Fantastic. So now our second step is we need to test our statistic. We need to pay attention and solve for the test statistic. Which means that as we should recall, F is equal to 950 divided by 890, which is equal to 1.0674 when we round to four decimal places. We now for our third step need to solve for the degrees of freedom, so we can state that DF1. So, DF1 degrees of freedom is equal to 8. And that's for our first sample population. So DF1 is equal to 8, and DF degrees of freedom for our second population will be equal to 7. Our fourth step now is we need to determine the critical value where we have a level of significance alpha is equal to 0.025, which will mean that our critical value F subscript 0.025.87 will be approximately equal to 4.89. Fantastic. So now we need to do our comparison for our fifth step, so we need to take 1.0674, and we need to note that this is less than 4.89. So because 1.0674 is less than 4.89, we can make our final conclusions as such. We do not, I'll put an X next to age 0, we do not reject the null hypothesis on an X next to null hypothesis, so we do not reject the null hypothesis. There is not enough evidence that sigma 1 squared is greater than sigma 2 squad. So that will mean our final answer without a doubt, will have to be multiple choice answer letter B. Which once again is, 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

Level of Significance (α)

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