Testing the Difference Between Two Means (b) find the critical...

Question

Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.

Interval Training

A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

Table comparing maximum aerobic speed of athletes before and after sprint interval training sessions.

Accepted Answer

Hi everybody. Here is our next problem. A researcher is testing whether a sleep tracking device improves sleep quality scores. Sleep scores are measured before and after using the device on 10 individuals. The test is right tailed, and the significance level is alpha equals 0.10. What is the critical T value and what is the rejection region? So let's think about first, what are null and alternative hypotheses. So, we want to know whether the device improves scores, and our scores are measured before and after. So we use a paired. Tea test. Those are two samples of the same individuals before and after, and we're looking at the difference. So after using the device, And subtracting the sleep score before using a device. And we want to see it improve sleep scores, which would mean we'd want to see that difference being greater than 0, being positive. I'll say that's an improvement. So to formulate our hypotheses, recall that our null hypothesis, we want to have a statement of equality, we want it to be a status quo. So that would be that our mean difference. Is equal to 0. While the alternative hypothesis would be that the mean difference is greater than 0. And as we know, that tells us we have this right-tailed test. So, always a good idea to sketch that graph. To think about what we're looking for, so we've got our distribution curve. We have a right tailed test, so we're looking for a critical T value. And our rejection region is going to be that region that I'll highlight in blue, to the right, or under the curve to the right of our critical T value. And that blue area is going to have that blue region is going to have an area, alpha equal to 0.10. So we're going to compare our T score that we calculate. With that critical T score, and we're going to reject. Our null hypothesis, if our calculated T value is. Greater than our criticality value. So, we use a table of T values, and we look up a one-tailed test. Where alphas equal to 0.10. And then we always need degrees of freedom for this test, which equals N minus 1. We are told we have a sample of 10 individuals, so that's going to equal 10 minus 1 or 9. So, 9 degrees of freedom, alpha equaling 0.10. And from the table, that's going to give us a critical T value of 1.383. So, now we compare, so that's our critical T value which we're asked for. And then our rejection region, remember is that blue area there. So it means we're going to Reject null hypothesis, if the Tval we come up with is greater than our critical T-value, so greater than 1.383. So, again, we were given an example of wanting to know whether a sleep tracking device improved sleep scores that were measured before and after using the device on 10 individuals. We have a right tilt test with alpha equals 0.10, and our question asks us for the critical T-value and the rejection region. And we get a critical T-value from the table of 1.383. Our rejection region, we reject our hypothesis, if our T value from our experiment is greater than 1.383. See you in the next video.

Key Concepts

Dependent Samples

Hypothesis Testing

Critical Value and Rejection Region

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