V10 Sampling Distribution Theory

This is part five of our video series on Sampling Distribution Theory. I'm Renee Clark from the Swanson School of Engineering at Pitt. In this video we're going to talk about the topic of the standard normal, or Z, probability table that's found in the back of statistics textbooks, as well as examples of using this table.

Okay, this is an example of a table of probabilities, or areas, whichever way you want to look at it, under the standard normal, or z-curve. Okay, and a table such as this is found in the back of most inferential statistics textbooks because a table such as this is used extensively in calculating confidence intervals or running a hypothesis test (or other inferential statistical techniques). Okay, so the use of these tables is quite important. Okay, so, the first thing that we want to notice is the following. I want to point out the visual legend to you, okay? So, the visual legend will show you a picture of the… the… the probability dist- distribution, or the curve we are considering. In this case, it's the normal standard, normal z-curve, okay? So, along the x-axis sits the actual Z value. Okay, you can see that it's small but you can see it. Okay, this Z value is a combination of the row and column headings. So, what I have circled in red are row headings. What I have circled in blue are column headings. Okay, now just as a quick review recall that Z is the number of standard deviations, okay, above or the below the mean of zero.

So, here is that mean of zero. Okay. That's just by way of review. Now, in order to use this table, the first thing you want to remember is that the Z value, which is formed by a combination of the row and column headings, sits along the x-axis. So, Z… the Z… Z value sits along the x-axis of that curve. The numbers in the middle of the table. So, all these, okay, they are probabilities or areas under our standard normal or Z curve.

Okay, and, in fact, based on the visual legend, okay, in this case they are probabilities to the left of the Z value, which sits along the x-axis. Okay, so, let's just delete that. Right there, as you can see from the legend, the area that is highlight… highlighted is actually to the left of that z value. You can see that. Okay, so for this table, these are probability… probabilities or areas that sit to the left of the particular z value along the x-axis.

Okay, now, in the back of your book, there are two pages that contain areas or probabilities under the standard normal curve. Okay, now, the first page is for values of Z that are negative, as you can see here going to circle the… the… the row header there. Okay, so values of Z that are negative. Okay, so they are values of Z that sit zero and to the left. Okay, so, from this table you can see that Z is as small as -3.49, okay, and that's formed by a combination of the row header and the column header, okay?

Okay, keep in mind that the prob… probabilities that are shown in these tables are to the left, or less than the particular Z value. And, again, can tell that just by looking at the visual legend, it shows the Z value and then it shows the Shaded area to the left of it or less than it. Okay, so, using this table, what is the probability Z less than 2.51? Okay, so in order to determine this, we go down to -2.5, okay, and then we go across to .001, which sits in the 100th place, okay? And then, from there, we read off the probability that sits at the intersection. So, the answer to this question is .60, okay?

What if we want to find out what is the probability, Z, less than - 3.17. We're going to do the same thing… same thing. We go down to… we find Z equal 3.1. Okay, then we go across to 007 or 7 in the hundreds place, okay, and we read off the probability or area that's at the intersection of those two. - 3.17 is right there, - 3.17.

So, therefore, the probability less… Z less than - 3.17 is 0.8. Very small probability, okay? Let's try one more.

How about the probability Z less than 0.4. Okay, we're going to go down to here 0.0. Then, we're going to go over to 0.004 or 4 in the hundreds, and we are going to read off the probability value that sits at the intersection of those two, which is right there. Okay, so the answer to this question is .4840.

Okay, okay. Thank you to the National Science Foundation for supporting our work under Grant 233582. Thank you for watching.