yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Sampling distribution of sample proportion part 2 | AP Statistics | Khan Academy


3m read
·Nov 11, 2024

This right over here is a scratch pad on Khan Academy created by Khan Academy user Charlotte Allen. What you see here is a simulation that allows us to keep sampling from our gumball machine and start approximating the sampling distribution of the sample proportion.

So, her simulation focuses on green gumballs, but we talked about yellow before. In the yellow gumballs, we said 60 were yellow, so let's make 60 percent here green. Then let's take samples of 10, just like we did before, and then let's just start with one sample.

So, we're going to draw one sample, and what we want to show is we want to show the percentages, which is the proportion of each sample that are green. So, if we draw that first sample, notice out of the 10, 5 ended up being green, and then it plotted that right over here under 50 percent. We have one situation where 50 were green.

Now let's do another sample. So, this sample 60 are green, and so let's keep going. Let's draw another sample, and now that one we have, we have 50 are green. So, notice now we see here on this distribution two of them had 50 green. We could keep drawing samples, and let's just really increase, so we're going to do 50 samples of 10 at a time.

So here we can quickly get to a fairly large number of samples, and here we're over a thousand samples. What's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6.

We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. As we draw more and more samples, we should get even closer and closer to those values, and we see that for the most part we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before.

Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. That makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable.

But what if our population proportion is closer to zero? So, let's say our population proportion is 10, 0.1. What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10, and so you can imagine that the distribution is going to be right skewed. But let's actually see that.

So here we see that our distribution is indeed right skewed, and that makes sense because you can only get values from 0 to 1. If your mean is closer to zero, then you're going to see the meat of your distribution here, and then you're going to see a long tail to the right, which creates that right skew.

If your population proportion was close to one, well, you can imagine the opposite is going to happen. You're going to end up with a left skew, and we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation.

So, let's do a population proportion that is right in between. So here this is similar to what we saw before; this is looking roughly normal. But now, and that's when we had a sample size of 10, but what if we have a sample size of 50 every time?

Well, notice now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution. The reason why that made sense, the standard deviation of your sample proportion is inversely proportional to the square root of n, and so that makes sense.

So hopefully, you have a good intuition now for the sample proportion, its distribution, the sampling distribution of the sample proportion, that you can calculate its mean and its standard deviation, and you feel good about it because we saw it in a simulation.

More Articles

View All
Sal Khan's thoughts on mastery learning
This idea of mastery learning was always kind of this gold standard. This was actually as a part of a fellowship I had while I was at MIT called the Eleranta fellowship to make a learning software for students with ADHD. It immediately struck a chord with…
Derivatives of inverse functions | Advanced derivatives | AP Calculus AB | Khan Academy
So let’s say I have two functions that are the inverse of each other. So I have f of x, and then I also have g of x, which is equal to the inverse of f of x, and f of x would be the inverse of g of x as well. If the notion of an inverse function is comple…
When to Launch Your Startup and When to Wait
I think this is the image founders have of the launch, which is it’s going to be like the launch, and it’s going to be like the Oscar ceremony or something, where there’s just going to be like hordes of people. And like you’re going to be treated like a c…
Watch Shopping With An Unlimited Budget With Teddy Baldassarre
And then of course, I crushed you like the cockroach you are. I just took off. I was a chicken once, now I’m a cockroach. I’m changing. Here I am, one-zero already. I don’t agree. You know, Teddy, the funny thing is, you’re living with this inventory ever…
Sex Myths | Original Sin: Sex
After 100 years of sex education appearing in schools around the globe, young people are more confused than ever. Many blame this on political agendas, which they believe stand in the way of a student’s right to know. It’s profoundly shocking that you wou…
Enrique Iglesias - Bailando ft. Descemer Bueno, Gente De Zona
I look at you, and it takes my breath away. When you look at me, my heart skips a beat. My heart slowly beats. In the silence, your look says a thousand words (uh). The night when I’m begging you not to let the sun come up. Dancing (dancing), dancing (da…