V28 Hypothesis Testing

Welcome to part four of our video series in support of hypothesis testing. In this video, we are going to discuss use of the T distribution for performing inference on the difference in two means from independent populations. I'm Renee Clark from this Swanson School of Engineering at the University of Pittsburgh.

Okay, so, when we are doing inference on the difference in two means, mu 1 minus mu 2, from independent populations, okay, in the case where our population variances are unknown, okay, so our Sigma 1^ squared and our Sigma 2^ squared are both unknown, and, in addition, our sample sizes out of the population are small, or less than 30, what do we do? In that case, well, we recall that we must use the T distribution in this case for inference, okay? Okay, now, when our variances are unknown, or our population variances are unknown, but we have reason to believe that they are equal or that that spread of the two populations is equal, okay, what we can use in that case to get a better estimate of the variance is what's known as the pooled estimate of the variance, okay? That's given by the symbol S Sub p squared, and what the pooled estimate does is it brings together your two sample variances, S1 squared and S2 squared… it pulls them or combines them using their degrees of freedom as weights. Okay.

Okay, so let's then talk about the case of performing inference on the difference in two means. We have independent populations, okay, but we don't know our population variances. We've got small samples, however, we have a reason to believe that our population variances are equal. Okay, so, we're going to use the T distribution. Here is our test statistic, or our… our T random variable. Okay, in order to use T, okay, it must be the case that each of the underlying populations is normally distributed. So, X1 must be normally distributed and X2 must be normally distribut… distributed. Okay, the central limit theorem does not play a role in the T distribution. Okay, but, again, the case of where we have reason to believe that the sigmas or Sigma squares are equal, we can use the pooled estimate of the variance, sp^ 2.

Okay, so, you'll notice in our test statistic we have SP in the denominator. Okay, our degrees of freedom are given by N1 + N2 minus 2. Okay, in calling your attention to our null hypothesis, this is our typical null hypothesis mu1 equal mu 2 that can be rewritten as mu1 minus mu 2 is zero, okay, or our hypothesized difference is zero. Okay, we then insert that zero into the test statistic equation for the hypothesized difference, and, again, just keeping in mind that this D Sub 0 equal mu1 minus mu 2. Okay, but, when we insert that zero, that term vanishes.

Okay, so, let's talk about the opposite case. Again, still performing inference on the difference in two means, independent populations, we still don't know our sigmas, we have small samples, and, in this case, we have reason to believe that the sigmas are not equal, or we have… we do not have reason to believe that they are equal. So, in this case, we cannot use the pooled estimate of the variance, okay? We don't want to use the pooled estimate of the variance, okay? It doesn't make sense to pool those sample variances when we don't have reason to believe that the sigmas are equal.

Okay, in this case, we use T… still using the T distribution… but you'll notice in the denominator there is no s sub p. There's no pooled standard deviation. Rather, the sample variances are given individually. Okay, but, in using T, our underlying population for X1 as well as X2 must be normally distributed. The central limit theorem does not play a role, okay? The degrees of freedom, in this case, is calculated in that very messy formula (which we won't be doing any such calculations). That's the degrees of freedom, okay, but if you were to do these calculations, likely you're going to get a non-whole number, something like degrees of freedom is 9.54 is possible, right? So, if you were to do… if you were to calculate these degrees of freedom, you would round down to the largest whole number if the value that you get has a… has a decimal component. So, in this case, if we were to get 9.54, we would say, in this case, our degrees of freedom is 9. Okay, but, again, null hypothesis mu 1 equal mu 2 is the typical null hypothesis. Another way to say that is our hypothesized difference of D Sub 0 equal mu1 – mu 2 = 0. To proceed with a proof by contradiction, we insert that zero into the test statistic, which that term then vanishes.

So, finally in this video, I wanted to discuss the relationship to the confidence interval. Okay, so, we're doing inference on the difference in two means and we have the following typical null hypothesis, okay, mu 1 equal mu 2. Okay, which another way to write that is that the hypothesized difference equal to mu 1 minus mu 2 is zero, right? So, if we're hypothesizing that they're equal, we're essentially hypothesizing no difference between the two means. Okay, now, this general relationship applies to any distribution you're using whether it be T, we're talking about t in this… in this video, but also applies to use of the Z distribution. Okay, so, let's look at an example. Let's say, for a given hypothesis test that we're running, we happen to calculate an associated confidence interval of 2.95 to 3.65, okay, and it is a confidence interval on the difference between mu 1 and… and mu 2. Okay, so, it… there's some lower limit and some upper limit that we calculate. Okay, as you'll notice in this confidence interval that I have starred in blue, 0 is not contained in that interval, right? 0 is less than the 2.95. Okay, so, if 0 is not in this particular interval, 0 or the hypothesized difference of 0 is not a plausible value for the difference in the mean. So, if it's not plausible, we reject zero as plausible for mu 1 minus mu 2, and, ultimately, what we end up doing is rejecting the null hypothesis in this case.

Okay, as a second example, let's say I calculate a confidence interval in this test of -1.25 to 2.25. Okay, I'll star this one in green, okay? So, for the confidence interval starred in green, you'll see that zero is contained in that confidence interval, right, because the lower limit is negative, the upper limit is positive. So, that confidence interval crosses over 0. So, in this case, 0 is a plausible value for the difference in the means for mu 1 and mu 2. So, we're certainly going to not reject ,or we're going to fail to reject, zero as plausible for the difference in the means. So, ultimately, in this case, we would fail to reject the null hypothesis.

We wish to thank the National Science Foundation under Grant 233582 for supporting our work. Thank you for watching.