V24 Estimation

Welcome to part seven of our video series related to estimation. In this video, we are going to review the Chi Square distribution and statistic as well as the F distribution and statistic. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, so, first, the chi square distribution. Okay, this distribution, as you'll recall, is used to investigate or estimate sigma squared, which is our population variance. Okay, the Chi Square distribution, of course, is always… is also used to estimate Sigma, okay, which is the population standard deviation. So, either one… can look at it either way, okay, and this is done by taking a sample from a normal or normally distributed population.

Okay, important point about using the Chi Square distribution to make estimates is that your population must be normally distributed. Okay, so, in other words, the central limit theorem does not have a role in the Chi Square distribution. The point estimate of Sigma squared is s squared, or the sample variance. Okay, you'll also recall from earlier in our video series that this particular statistic here, or the sample size minus one times the ratio of the sample variance to the population variance, has a Chi Squared Distribution, we write Chi square like that, and it has n minus 1 degrees of freedom. You recall that, in general, the Chi square distribution is a continuous distribution. It's skewed to the right, it's not symmetric, and the convention for using it, including for the back of the book, is that alpha represents the area to the right under the Chi Square curve… to the right of the critical value Chi Square sub alpha.

Okay, again, this should look familiar to you. This is the critical values of the chi Squared Distribution from the back of the book. Okay, here's a visual legend that you should always reference when using a table. You'll see, again, alpha is that area to the right… little bit difficult to say… to see, but critical value sits along the x axis is Chi squared sub alpha degrees of freedom. Down the left, values of alpha. Across the top, there are two pages of the…of these critical values for the Chi Square distribution in the back of the book. Values of Chi Square sit in the middle of the table, okay?

Okay, now the F distribution. Okay, the F distribution is used to investigate or estimate the ratio of the variances from two independent populations, okay, which is given by the ratio of Sigma 1^ 2 to Sigma 2^ 2. Okay, and, of course, the F distribution is also used to investigate or estimate the ratio of the standard deviations from the two independent populations. You can look at it either way. Okay, this is done by taking samples from two independent, normally distributed populations.

Okay, again, an important point about the F distribution, just as with the… the chi Square distribution, your populations must be normally distributed in order to use the F distribution. Okay, there is not a concept of the central limit theorem with the F distribution. Okay, the point estimate for the ratio of the population variances is going to be, of course, the ratio of the two sample variances, S1^2, S2^2.

Okay, you'll recall that this statistic right here is distributed according to the F distribution. Okay, so, that statistic is in the numerator… has a numerator of… for the first population, the ratio of the sample variance to the population variance. Okay, and then, in the denominator, the ratio from the… for the second population, the ratio of the sample variance to the population variance. Okay, and so, this particular statistic, here, can also be rewritten in this way just by rearranging some terms. Okay, the… the… there are two sets of…there are two degrees of freedom associated with the F distribution: from the first population, N1 minus one, and from the second population, N2 minus 1. So, each sample size minus one. Recall that the F distribution is also a continuous distribution like the chi squared, also skewed right, not symmetric. Okay, the convention in…including… for the table in the back of the book is alpha represents area to the right… under the F curve… to the right of the critical value for f that sits along the x-axis. We call that F sub Alpha, so that Alpha is area to the right. Okay, this should look familiar to you. This is the table of critical values of the F distribution that's in the back of the book. There are four… four total pages of these critical values in the back of the book. Across the top is your nu one, or N1 -1 (first degree of Freedom) Second degree of Freedom, nu 2 sits down the left hand side. That is your N2 minus one.

Okay, critical values for f sit in the middle of the table. These are your F… it's… sub Alpha. This particular table is for Alpha of .05 because of that subscript, there. Okay, and, again, always take note of your visual legend when you're using a table. F sub Alpha sits right there, do that in blue. F sub Alpha- a little difficult to see there, but F sub Alpha is your critical value. It sits along the x-axis. Alpha, of course, by convention, you… in looking… in the visual legend represents your area to the right under the curve… curve to the right of your F sub Alpha. Okay, we also have tables in the back of the book for an alpha of .01 as well.

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