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.

Annual Salaries An employment information service claims that the standard deviation of the annual salaries for public relations managers is less in Louisiana than in Florida. You select a sample of public relations managers from each state. The results of each survey are shown in the figure. At α=0.05, can you support the service’s claim? (Adapted from America’s Career InfoNet)

Accepted Answer

Hi everybody and welcome back. Our next question says, a company wants to know if the standard deviation of delivery times is less for its new warehouse, warehouse X, than its old warehouse warehouse Y. For warehouse X, the sample standard deviation. S1 S1 equals 3.8 hours while sample size and sub 1 is 16. For warehouse Y, sample sample deviation S sub 2 is 5.1 hours, while sample size and sub 2 is 18. At alpha equals 0.05, is there evidence that the new warehouse has less variability? No, there is insufficient evidence that the new warehouse has less variability each subzero. B not enough information. The test is right tailed. Yes, there is sufficient evidence. The new warehouse has less variability. So our no hypothesis here would be. That the variability of sample 1 is equal to the variability of sample 2. While our alternative hypothesis would be that the variability of sample one, which is the warehouse X or new warehouse. is less than the variability of sample 2. With that lesson we're talking about a left. Tailed Situation here. So we need to be especially alert in this case. Our test statistic. Is F, which we calculate as. The variability of sample one, so as one squared. Divided by the variability of sample 2, so S 2 squared. So that's going to be 3.8 squared, as that's the variability of of uh sample one. Divided by 5.1 squared. Remember that we have to square the standard deviation since we're talking about variability. And that will equal Approximately 0.555. So, to look up our critical value, we need to think about our degrees of freedom. So we look at the degrees of freedom in the numerator. Which is of course sample one. So that will be N minus 1. And sub 1 minus 1, we should say, which equals 16 minus 1. So 15. And the degrees of freedom in the denominator. Which is N2 minus 1. Which equals 18 minus 1, so 17. And then we recall that our alpha is 0.05. Now, here's where it gets a bit tricky to find a left-tailed critical F value. We need to switch the degrees of freedom. So instead of looking at 1517 numerator denominator. We are going to, so we'll say left. Tailed critical value. We will treat the degrees of freedom of the denominator, let's put D. As 15 and the degrees of freedom as the of the numerator as 17, so those are the values we will take as we look it up. So using those two degrees of freedom and our alpha equals 0.05. The value in the cell at that point is going to be. 2.38. However, left tailed critical value. So our critical value is actually going to be the reciprocal. So 1 divided by 2.38. So it'll be approximately 0.420. So, that's definitely something you have to be really careful to keep straight when you're talking about F. Values there. So, now we compare our left-tailed critical value of 0.420 to our test statistic of F. 0.555 is greater than 0.420. It's our critical value Since this is left-tailed, that means we fail to reject. The null hypothesis. So, in that case, our answer here is going to be. Choice A No, There is insufficient evidence that the new warehouse has less, the new warehouse has less variability. When we look at our other answer choices, Choice B not enough information. We have enough information. Choice C, the test is right tailed, it is not, it is left-tailed. And Joyce D is incorrect, since we do not have sufficient evidence to reject that null hypothesis. So once again we were comparing the standard deviation of two samples. And wanting to see if the standard deviation of our sample one was, or the variance, excuse me, of our sample 1 was less than the variance of our sample two. So using a test statistic of F, and we found our left tailed critical value of F. Our test statistic was greater than our critical value, and therefore we failed to reject our null hypothesis. So our answer again was choice A, no, there is insufficient evidence that the new warehouse has less variability. See you in the next video.

Key Concepts

Hypothesis Testing

F-Test

Rejection Region

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