V11 Sampling Distribution Theory

This is part six of our video series in support of sampling distribution theory. I'm Renee Clark from the Swanson School of Engineering at Pitt.

In this video, we are going to do additional examples of using the Z probability table that's found in the back of most inferential statistics books. This is a direct continuation of part five in the video series.

Okay, in the back of your textbook, as I had indicated, there are two pages of probabilities under the Z curve, or the standard normal curve. In the prior video, we looked at values of Z that were negative. Okay, now we're going to use the page, page 736, that has positive values of Z.

Okay, so, if I draw in the legend here, these are values of Z that are to the right of the mean of zero. Okay, so, as you can see from circling the row headers, these are positive values of Z. The column headers are the same as… as in the previous video. From this table, you can tell that Z is as large as 3.49. Okay, but, on this table, the probabilities are still those to the left of the z value even though there is not a legend shown on this page. The legend is the same. It’s showing probabilities that are to the left of the Z value. Okay, so, if we wanted to determine the probability Z less than 1.96, we would do the same thing. As previously, we would find 1.9 in the table. Okay, we would find 06 column header, and we would read off the probability that's at the intersection of those, which is .975.

So, the probability Z less than 1.96 is .9750. Okay, now, important related question: What if we wanted to know the probability Z greater than 1.96? So, the… we just determined the probability Z less than 1.96. What if we want probability Z greater than 1.96? Very simple to determine. You will remember that, under the normal curve, any normal curve, the total area, or the total probability under that curve, must equal one. Okay, so, if we want the probability Z greater than 1.96, it's equal to 1, use the probability Z less than 1.96, right? Because we were to draw this in here.

Here's 1.96. Okay, the probability… ility that Z is less than 1.96, which is in green, plus the probability Z greater than 1.96, which is in blue, must equal one. Okay, so this will be 1 minus probability Z less than 1.96, which we determined just above to be .975. And when we do the math on that, that's equal to 0.025. So, that is the answer to this question. So, the area in blue here, .025, area in green determined initially .9750.

Okay, let's do one additional example, which is something that you will likely be asked to do frequently, is to determine that… the probability that Z is between two numbers. Okay, so, for example, in this case, we're being asked to determine the probability that Z is between 1.25 and 2.54. Okay, with this type of a problem, I always start by drawing a picture. It's very, very helpful. So, Z curve- mean of zero. We're being asked to determine that Z, which sits along the x-axis, is between 1.25 and 2.54. Okay, so, it's actually the area that's shaded in yellow under the curve. Okay, can you see that, if you take all the area under Z to the left of 2.54 in… in blue, and you subtract off all the area to the left under the curve to the left of 1.25, which is in purple, what you're left with is the area in between, which is the yellow shaded area? Okay, so, therefore, if we want to find the probability that Z is between 1.25 and 2.54, or, another way to write that is the probability that 1.25 less than Z less than 2.54, we simply take the probability that Z less than 2.54, or all the probability to the left of 2.54. We subtract off all the area, or probability, that Z is to the left of 1.25, just like I showed in… above. Okay, to get these values, we simply can read them off the table. Okay, 2.54. The value at the intersection there is .9945.

So, 0.9945. Sub… subtract off probability Z less than 1.25. Remember, these probabilities that are shown in these tables are areas to the left of Z. Okay, probability Z less than 1.25…1.25. The probability at the intersection of those two is right there- .8944. 0.8944. And, when you subtract those two numbers, you come up with 0.101. Okay, that's the final answer. So, the probability that Z is between 1.25 and 2.54 is 0.11.

Thank you to the National Science Foundation for supporting our work under Grant 2335802. Thank you for watching.