Sampling distribution of the difference in sample means: Probability example

- [Instructor] In the previous video, we began our journey trying to find the probability that the mean weights from the samples are more than six grams apart from each other. And to do that, we started studying or we started thinking about the sampling distribution of the difference of sample means. And we figured out the mean of that sampling distribution. We figured out the standard deviation of that sampling distribution, and we were able to establish that it is normal, so that the sampling distribution of the difference of sample well means will look something like this. So it is normal. So it is going to have that classic bell shape like that. We know it's mean is at five grams. So this is five grams. And we know that the standard deviation is roughly 0.79 grams. So this is one standard deviation above, maybe two standard deviations three. This is one standard deviation below, two standard deviations below, three standard deviations below, this right over here would be 5.79 grams. This over here would be 4.21 grams. Now, if we want to find the probability that the mean weights from the samples are more than six grams apart. Well, remember this is the difference between the sample mean from A and the sample mean from B. And so there's one situation where A is more than six grams larger than B, or their sample mean from A is more than six grams larger than the sample mean from B. And that would be this area right over here. Six is right around here. It would be that area, but there's also a possibility that the sample mean from B will be larger, will be more than six grams larger than the sample mean from A, so if you really were to extend this far to the left, and you're really not gonna be able to see it much like this. So this is five, maybe this gets us to about zero right over here, and then you're gonna get negative six someplace out here. There is some area under the curve where this difference is more negative than negative six or, in other words, think about it's less than negative six. So to figure out this probability, we to calculate both of these areas. Now you could do that with a Z table. And we've done that in many examples before. Or we could use some type of online tool or all online calculator. So this is at a site stapplet.com/normal.html. And what we can do is we know we have a normal distribution that has a mean of five grams, a standard deviation of 0.79. And then we can plot the distribution. There we have there, and then we can calculate the area. Well, we could calculate the area outside of a region. And then that region, the left boundary, would be negative six. And then the right boundary would be six. And then if we calculate that area, it is, you see the right boundary right over here or the part that's greater than the right boundary. But there's a little bit of be negligible, that's far to the left that we're not seeing as well, but there you have it: that area which is mostly the one that you're seeing here visualize, you're not seeing the one on the left, it is 10.28% the combination of the two. And so that's our probability, our probability, we just found it. It is 10.28%, and we're done.