V32 Hypothesis Testing

Welcome to part eight of our video series in support of hypothesis testing. In this video, we are going to review the chi square test statistic involving the population variance, sigma squared. We're going to do a visual review of the chi square distribution. We will also review the F test statistic involving the ratio of two population variances from independent populations, and also do a visual review of the F distribution. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, for inference on one variance, sigma squared, okay, we know that a point estimate for sigma squared is s squared, or the sample variance. Okay, so, if we take a sample out of a normally distributed population, then this statistic or random variable, which is n minus 1 * the sample variance over the population variance, has a chi squared distribution, which we write as chi squared with n minus 1 degrees of freedom. Okay, or sample size minus one degrees of freedom.

Note that our population must be normally distributed when using the Chi Square distribution for this inference. Okay, let's do a visual review of the chi square distribution. Okay, recall that the chi square distribution, as shown here, is… it's a continuous distribution. It is skewed to the right because it has a long right tail to the right. Okay, if I were to identify this point right here along the x-axis, that's a particular critical value of the chi square distribution that is actually labeled as chi^ 2 sub alpha / 2. The reason being is because this particular critical value is showed as having alpha over 2 area to the right in the picture, and remember that the subscript on the critical value by convention refers to area to the right of it under the curve. Okay, so, likewise… wise, this value right here in red along the x-axis is designated as chi^ 2 sub 1 - alpha / 2, and the reason for that is you will notice that, in the picture, it has alpha over 2 area to the left of it under the curve. That means it must have area 1 - alpha /2 to the right, since total area under the curve must equal to one.

Okay, so, when we are doing inference with one variance, this is our null hypothesis versus the two-sided alternative that the variance is not equal to some hypothesized variance. Okay, but, to proceed with a proof by contradiction in order to prove the alternative, we have to assume the null to be true. Therefore, we will insert the null hypothesized value for sigma squared into our chi square test statistic. Okay, and, this quantity becomes our chi square random variable or test statistic for performing inference on the variance, sigma squared. So, we have a hypothesized value for sigma squared in the denominator.

Okay, let's next… next talk about inference for the ratio of two variances from two independent populations, or sigma 1^ 2 over Sigma 2^ 2. Okay, so, we know a point estimator. From the past, we know the point estimator for sigma 1^ 2/ sigma 2^ 2 is s1 ^2 / s 2^2, okay, or the ratio of the two sample variances from those populations. Okay, so, if we take a sample from each of two normally distributed independent populations, okay, then this particular statistic here, okay, which is the ratio of the sample to the population variance in the first pop… population, divided by the ratio of the sample variance to the population variance of the second population. That particular random variable or statistic has an F distribution, right, and this can actually be written in that way as well just by some simple rearrangement of the… of the terms. Okay, this random variable..able has n1 -1 and n2 -1 degrees of freedom- so two different degrees of freedom. But, like the chi square distribution, each population, from which… from which we're sampling to do this inference, must be normally distributed. So, when using the F distribution in this way, your populations must be normally distributed.

Okay, so, again, let's do a visual review of the F distribution. This point right here along the x-axis, which is a particular critical value of the F distribution, would be labeled, or should be labeled as F sub alpha / 2 because it is shown as having area of alpha over 2 to the right, and you'll remember the subscript on the critical value by convention represents area to the right. Okay, so, likewise, this particular critical value of F at that point along the x axis should be labeled as F sub 1 minus alpha / 2. Okay, the reason being is because it is shown as having alpha over 2 to the left of it under the curve. So, if it's got Alpha over 2 area to the left, that means it must have 1 - alpha over 2 area to the right of it under the curve, right, because the total area under the F curve must equal one.

Okay, in performing inference for the ratio of two variances, sigma 1^ 2 to sigma 2^ 2, okay, this is our typical null hypothesis- that the two population variances are equal. Okay, another way to write that is that the ratio of sigma 1^ 2 to sigma 2^ 2 equal to 1, simply by dividing by sigma 2^ 2 on each side of the equality. Or, equivalently, we could say… we could write this as sigma 2^ 2 to sigma 1^ 2 equal to 1. Either way, in which case you would be dividing by sigma 1^2 on each side of the equality. Okay, so, in proceeding with a proof by contradiction, what we are going to do is we are going to insert the null hypothesized information into our test statistic. Okay, so, we saw on the previous page that this is our test statistic distributed according to the F distribution involving our two population variances. Okay, it can be written either way, but what we're going to do is, when we substitute this information in about the ratio of the two population variances, we're going to get a cancellation, okay? If that hypothesized ratio is one that's going to cancel or, alternatively, if you're going to look at it the other way, it's going to cancel. So, what you're ultimately left with is that the F test statistic can be simplified to the ratio of the sample variances, s1^ 2 over s2 ^2- probably the simplest test statistic we've seen thus far. Okay, this, of course, will have n1 -1 degrees of freedom and n2 - 1 degrees of freedom- two separate degrees of freedom.

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