V9 Sampling Distribution Theory

Welcome to our fourth video in the series on sampling distribution theory. I'm Renee Clark from the Swanson School of Engineering. In this video, we are going to talk about the Z transform, or the Z transformation, of a normal random variable and why we transform normal random variables to Z. Okay, so a z transformation, or transform, is used to standardize any normal random variable which we will call X. Okay, this standardized variable is then called Z. Okay, so, Z is obtained by taking X, subtracting off its mean- that's the population mean- dividing by its standard deviation (this is the population standard deviation). Okay, Z then has what's called a standard normal distribution. Okay, in other words, Z is distributed normally. Okay, but it will have a mean of zero and a variance of one, okay?

Okay. That's why we call it standard because it has a mean of zero and a variance of one. As a side note, Z is actually the number of standard deviations, or sigmas, okay, that X is above or below the mean value of zero. Okay, because you're dividing by the value of the standard deviation. That's why Z is the number of standard deviations that X is above or below the mean value of zero. Okay, why do we transform? Okay, the answer to this question is that this enables us to avoid having to integrate this PDF of the normal distribution, which you have seen before, which is quite a complicated PDF. And why would we have to integrate this PDF in order to calculate probabilities or areas under a normal curve? We need these probabilities and areas in order to do inferential statistics.

Okay, so what has already been done is probabilities, or areas (same thing), have already been calculated and tabulated such as shown here for the standardized normal variable Z. Okay, which has a mean of zero and a variance of one. Okay, so, for example, the table in the back of your book has probabilities under the standard normal curve, or the Z normal curve, when mu is equal to Z and the variance is one. Okay, that's why we would standardize any normal random variable X to be a z. Okay, because those probabilities have already been tabulated and we can use that table in order to calculate probabilities. Do you see that it would be impossible to create a table in the back of a book for every possible mu and sigma? You see how mu and sigma are input variables into this PDF or probability calculation, okay? That's why it's important that we have to standardize them, okay, to 0 and one. Okay, so that we have one table based on just one value of mu, which is zero, and one value of Sigma, which is one. Because it would be impossible to have tables for every combination of mu and sigma that could exist.

We thank the National Science Foundation under Grant 233 582 for supporting our work. Thank you for watching.