Consider our example again, suppose that n was 16 rather than 100. Then our test statistic remains the same. It's the sample mean minus the hypothesis me, mean. Remember,we're testing H0 mu equal to 30. Versus Ha mu greater than 30. And then divided by the standard error of the mean where now, where we have square root 16 rather than square root 100, if you recall s was 10. This test statistic, how many estimated standard errors from the hypothesised mean, the sample mean is. Follows a t distribution with 15 degrees of freedom in this specific case. So under the null hypothesis the probability that is larger than the 95th percentile the t distribution is 5%. So we need to calculate that fifth percentile of the t distribution, this can be done with qt 0.95 and 15 degrees of freedom which works out to be 1.7531. Our test statistic, if we actually plug in the 10 in our x bar of 32, works out to be 0.8. And so we're failing to reject because 0.8 is smaller than 1.75. Let's also go through a two-sided test. Suppose that we wanted to re, reject the null hypothesis if in fact the mean was too large or too small. This doesn't make a lot of sense in our specific scientific setting, because we're specifically interested in whether or not thi, this particular population of obese subjects had a respiratory disturbance index, larger than 30 or reference value. However, it's often the case, that in scientific settings, a two-sided test is demanded, regardless of whether or not it makes scientific sense. So it's, it's important to understand how to do two-sided tests, and the fact that there are some instances where you are mandated to deal with two-sided tests, even though it doesn't necessarily make that much sense to consider the other side. So we want to reject, in this case, if we were to do a two-sided test, we want to reject whether or not mu is different from 30. So in other words, we'll reject if our test statistic is either too large or too small. In this case, because our test statistic is positive, we only need to consider the large side. What does change however, is in order to get that 5% in a way that allows our test statistic to be too large or too small, we need to split. The probability is 2.5% in either tail of the distribution, be it the t distribution for small sample sizes, or the z distribution for large sample sizes. So now, instead of qt at 0.95, we're going to get, do qt at 0.975 with, again, with 15 degrees of freedom. And so we want to reject if our test statistic is larger than this. Right, so let me draw an example of our t distribution. This will be the point qt 0.975, with 15 degrees of freedom would be 2.5% in that tail. And this is the point Q T .025 with 15 degrees of freedom. And that's 2.5% in that tail. That point right there. So we're going to reject if our test statistic is larger than this guy, or smaller than this guy. However, because the lower quintile is the negative of the positive quintile, we can always say that's the same thing as taking the absolute value of our test statistic and rejecting if it is too large. That point is made right here. So, in this case, we've failed to reject the one-sided test. And now in this slide we're showing that we failed to reject the two sided test. However, I think you'll probably already noticed, that because we've moved further out into the tails of t distribution with our rejection region, that if you fail to reject the one sided test that then you will also have failed to reject the two sided test. Usually we don't calculate the rejection region and perform the hypothesis test in the formal manner in which we've gone through in the slides by hand, instead, we usually pass the data to a function like t.test. And it outputs all the relevant statistics for us, for us to understand what's going on with the text. What's interesting is we already know how to do this, because we've used t.test in order to perform confidence intervals. We just haven't gone through the output to understand what it's doing for a test. Let's look at, in the using R package, the data father.son. And we'd like to test whether the population of son's height was equivalent to the populat, population mean of father's heights. Now, the observations here were paired. It was one son's measurement to one father's measurement, and so on. So its in this case we're going to take the difference and we want to test whether the difference in the heights is 0 or its non zero, we do that with t.test. And you can either pass the difference directly to the function t.test, or you could pass it, the two vectors, and then add the argument, paired equals true. It gives you your t statistic right here, 11.79. It gives you your degrees of freedom, right here, 1,077. So we had e, exactly 1,078 pairs that we took the difference of. Now, 11 is a quite large t statistic, so we reject the null hypothesis in this case. Also notice that the degrees of freedom are quite large so the distinction between a t-test and a z-test in this case is irrelevant. It very nicely gives us the t confidence interval, automatically when we do t.test. It's useful almost always to look at the confidence interval in addition to the output of the test. Simply because the confidence interval bridges this gap between statistical significance and practical significance quite nicely. You can see whether the range of values in the confidence interval are of practical significance or not, because it's expressed in the units of the data that you're interested in.
