Normal conditions for sampling distributions of sample proportions | AP Statistics | Khan Academy

What we're going to do in this video is think about under which conditions the sampling distribution of the sample proportions in which situations does it look roughly normal and under which situations does it look skewed right. So, it doesn't look something like this, and under which situations does it skewed look skewed left—maybe something like that.

The conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10. And if we take the sample size and we multiply it times 1 minus the population proportion, and that also is greater than or equal to 10. If both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape—the sampling distribution of the sample proportions.

So, with that in our minds, let's do some examples here. So, this first example says, "Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12 percent of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of over-ripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution of the daily proportions of over-ripe tangerines?"

Pause this video, think about what we just talked about, and see if you can answer this.

All right, so right over here we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe, is 12 percent. So, p is 0.12.

So, if we take n times p, what do we get? And p is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to 6, and this is less than or equal to 10.

So, this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is how is it going to be skewed? The key realization is remember the mean of the sample proportions of the sampling distribution of the sample of the sample proportions, or the mean of the sampling distribution of the daily proportions—that's going to be the same thing as our population proportion.

So, the mean is going to be 12 percent. So, if I were to draw it, let me see if I were to draw it right over here, where this is 50 percent and this is 100 percent, our mean is going to be right over here at 12 percent. And so you're going to have it really high over there, and then it's going to be skewed to the right. You're going to have a big long tail, so this is going to be skewed to the right.

Let's do another example. So, here we're told, "According to a Nielsen survey, radio reaches 88 of children each week. Suppose we took weekly random samples of n equals 125 children from this population and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches?"

Once again, pause this video and see if you can figure it out.

All right, well let's just figure out what n and p are. Our sample size here n is equal to 125, and our population proportion, that is the proportion that of children that are reached each week by radio, is 88. So, p is 0.88.

So now, let's calculate np. So n is 125 times p is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly—this is almost 90 of 125. This is actually going to be over 100, so it for sure is going to be greater than 10. So we meet this first condition.

But what about the second condition? We could take n 125 times 1 minus p, so this is times 0.12. So this is 12 of 125. Well, even 10 percent of 125 would be 12.5, so 12 is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it—I could just estimate it.

So, we meet that second condition. So, even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal.

One way to get the intuition for that is, so this is a proportion of zero, let's say this is fifty percent, and this is a hundred percent. So, our mean right over here is going to be at 0.88 for our sampling distribution of the sample proportions.

If we had a low sample size, then our standard deviation would be quite large, and so then you would end up with a left-skewed distribution. But we saw before the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up the standard deviation, and so it's going to look more normal. It's going to look closer to being normal, so we'll say approximately normal because it met our conditions for this rule of thumb.

Is it going to be perfectly normal? No. In fact, if we didn't have this rule of thumb to kind of draw the line, some might even argue that, well, we still have a longer tail to the left than we do to the right—maybe it's good to the left. But using this threshold, using this rule of thumb, which is the standard in statistics, this would be viewed as approximately normal.