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.

U.S. History Assessment Tests A state school administrator claims that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. A sample of 10 test scores from District 1 has a standard deviation of 30.9 points, and a sample of 13 test scores from District 2 has a standard deviation of 27.2 points. At α=0.01, can you reject the administrator’s claim? (Adapted from National Center for Education Statistics)

Accepted Answer

Hi everyone and welcome back. Our next question says An analyst is comparing the consistency of customer satisfaction scores between two branches of a retail store. A sample of 14 scores from branch X has a standard deviation of 18.7. And a sample of 11 scores from branch Y is a standard deviation of 22.3. At alpha equals 0.05, is there evidence to suggest the standard deviations are not equal? A fail to reject null hypothesis. There's no evidence that the standard deviations differ. reject null hypothesis. The standard deviations are significantly different. See fail to reject no hypothesis. The means are equal. Or the reject no hypothesis, the standard deviations are equal. So, in this case, because we're talking about standard deviation as our sample parameter. We're going to be thinking about comparing the variances of these two. So our hypothesis will be our null hypothesis will be. That the variance of sample one, so sigma squared sample one. is equal to the variance of sample 2, or sigma squared of sample 2. And then the alternative hypothesis will be. That the two variances do not equal each other. So, sigma squared of sample 1 does not equal sigma squared of sample 2. So when we're comparing variances, our test statistic is F. So test statistic. F Would be equal to the standard deviation squared, because we're talking about variant, so the variance. Of sample 1 divided by the variance of sample 2. So standard deviation squared, divided by standard deviation squared. So that will equal. If we put branch Y in the numerator, we have 22.3 squared. Divided by 4 branch X, 18.7 squared. And that's going to give us a value of 1.424 F. Now we need to think about our degrees of freedom. We have two samples. So, our numerator, branch Y degrees of freedom, so we'll put DF. Numerator That had 11 samples, so is 11 minus 1 equals 10. There are degrees of freedom in the denominator. Sput dinar In branch X, we had 14 scores, so that degrees of freedom will be 14 minus 1 equals 13. We have alpha equaling 0.05, but this is two-tailed. So we're remembering that the area. Under the tail in each tail. Is going to be alpha over 2. So that's what we're going to use when looking up our F value. So now we need our upper. Critical F value, since we're two-tailed. So when we look at that table, we need to look at our alpha, so alpha will be. 0.025 since it's the areas alpha divided by 2. And then we need to look at the Two degrees of freedom, so the numerator. Again, is 10 and the denominator 13. So, put 1013. So, when we look at that table, it's going to give us an upper critical value of 3.25, or we'll say approximately. 3.25. Now, the lower critical F value is a little trickier. So, still using alpha equals 0.025. We actually reverse the degrees of freedom. So we use the degrees of freedom from the denominator. And then numerator, we flip them. So we're going to use 1310. Instead, and then we calculate the reciprocal value of the F value that we get from the table. So that lower critical F value will be. The reciprocal one over, and the value you get from the table, we'll say equals approximately. Is 3.58. So, that's going to give us a lower critical value of approximately 0.28. So we know that our test statistic. The, if it's between those two critical values, it is, we're going to fail to reject the null hypothesis, it's outside, then it will be in the rejection region. So we know our test statistic was 1.42. And that is, when we look at our upper critical value, less than 3.25, or upper critical value. But it is greater than 0.28. Are a lower critical F value. So that means we fail to reject. The null hypothesis. So answer then will be choice A. We fail to reject the null hypothesis. There is no evidence that the standard deviations differ. So once again, we had 2 batch, 2 samples. One of 14 scores with a standard deviation of 18.7, 1 of 11 scores with the standard deviation of 22.3. At alpha equals 0.05, was there evidence to suggest the standard deviations are not equal? So, we were comparing variances, we got our test statistic of F, looked for our upper and lower critical F values. And since our test statistic F fell in between those critical values, our answer is choice A, that we fail to reject the null hypothesis, there's no evidence that the standard deviations differ. See you in the next video.

Key Concepts

Hypothesis Testing

F-Test

Critical Value and Rejection Region

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