Performing a Two-Sample F-Test In Exercises 1...

Question

Performing a Two-Sample F-Test In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Heart Transplant Waiting Times The table at the left shows a sample of the waiting times (in days) for a heart transplant for two age groups. At α=0.05, can you conclude that the variances of the waiting times differ between the two age groups? (Adapted from Organ Procurement and Transplantation Network)

Accepted Answer

Hello there. Today we're going to 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. A marketing firm conducted a survey of individuals who use smartphones for more than 4 hours daily. The table below shows the number of such users categorized by age group and type of battery usage. Standard batteries, long life batteries, age group 1, 402, 387, age group 2, 395, 390. Age group 3, 40. 10, 416, age group 4, 398409. Age group 5, 405, 420, age group 6, 400, 412. Perform a G2 test of independence at alpha equals 0.05 to determine whether primary smartphone battery usage is dependent on age group if G2 is equal to 0.841. Awesome. So it appears for this particular prompt we're asked to perform a G2 test of independence. Given that we have a level of significance, alpha is equal to 0.05 in order to determine whether the primary smartphone battery usage is dependent on the age group if G2d is equal to 0.841. So now that we know that we're ultimately trying to determine whether or not primary smartphone usage is dependent on age group, let's read off our multiple choice answers to see what our final answer might be. A is there is significant evidence that the type of smartphone battery usage is dependent on age group. There is insufficient evidence to conclude that the battery type of smartphone battery usage is dependent on age group. C is the test is inconclusive, and finally D is the test cannot be performed. Awesome. So it appears for this particular problem, our first step that we're going to need to take is we're going to need to state our hypotheses, starting with the null hypothesis which will denote as H0. So for our null hypothesis, this will be the type of smartphone battery usage being independent of age group. So once again, our null hypothesis H0 will be focused on the type of smartphone battery usage is independent of age group, whereas for our alternative hypothesis, which will denote as HA, and for our alternative hypothesis, this will be the type of smartphone battery usage is dependent on age group. So variances are different. And this will be, as we should be able to recognize, a two-tailed test. So once again, to quickly recap, our null hypothesis is focused on the type of smartphone battery usage is independent of age group, whereas our alternative hypothesis will be the type of smartphone battery usage is dependent on age group, and we need to note this is very important, that variances are different and that we need to recognize that this is a two-tailed test. So with that in mind, our second step now is we need to determine the degrees of freedom for this particular prompt. So we'll denote the degrees of freedom as DF and the degrees of freedom is equal to R minus 1 in parentheses multiplied by parenthesis C minus 1, which is equal to. When we plug in our known variables, parentheses 6 minus 1 multiplied by parentheses 2 minus 1, which is equal to 5 when we perform that calculation, or when we plug in parentheses 6 minus 1 multiplied by parentheses 2 minus 1 into our calculator, we will find that it's equal to 5. Our third step now is we need to determine the critical value. So from the T square table, which you should have a T square table in your textbook, or you can just quickly look one up online. So when we look at our T square table with a level of significance alpha is equal to 0.05, and we have a degrees of freedom, which is equal to 5, we will find that T squad subscript C for critical, so T squared critical will be equal to 11.070. And once again, this is when we're this is how we determine our critical value. So once again, T 2 C, which is our T squared critical value, is equal to 11.070. Fantastic. Going on to our fourth step now. Our fourth step is we need to make a decision, and we need to note that for our T2d calculation. We need to note that our G2d calculation is equal to 0.841, and this is just to make sure we don't get it confused with our G2 critical value that we found. So this G2d value is the value that's given to us by the prom itself. And once again, we're trying to make our decision here. So our T squared calculation is equal to 0.841, and this value is less than our T squared critical value, and because it is than our G2 critical value, which is equal to, once again, means our G2 critical value is bigger than our G2 value that's given to us by the problem itself because our critical G2 value once again is equal to 11.070. So because our critical G2d value is greater than the T2d value that's given to us by the problem itself, we fail to reject the null hypothesis, so I'll put an X next to H0, and because we fail to reject the null hypothesis, we can state that in conclusion, at the level of significance alpha is equal to 0.05. There is insufficient evidence to conclude that the type of smartphone battery usage is dependent on age group. So that will mean that our final answer when we review our multiple choice answers. Our final answer will have to be multiple choice answer letter B, which once again is there is insufficient evidence to conclude that the type of smartphone battery usage is dependent on age group. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Two-Sample F-Test for Variances

Hypothesis Testing Framework

Assumptions of the F-Test

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