Comparing simulated and theoretical sampling distributions - proportions

0:00- [Instructor] We're told that exactly 75% of a bowl of 1,000 marbles are blue. So exactly 750 of them are blue. Emanuel simulates a large number of samples of size n equals 200 taken from this population. The distribution of proportions of blue marbles from his simulation is shown.

0:19So what Emanuel is doing is each of these dots don't just represent a marble. They represent taking 200 marbles, a sample of 200 marbles, and then calculating the proportion of those 200 marbles that are blue and then plotting it in the appropriate bucket essentially.

0:38Is Emmanuel's simulation consistent with the theoretical sampling distribution of this population for samples of size 200? So pause the video, have a go at that. Okay, so first of all, what's the mean of our sampling distribution going to be, or at least our theoretical sampling distribution?

0:59Well, the mean of our theoretical sampling distribution is going to be the same thing as the population proportion. The population proportion here is 75%, or I could write that as 0.75. So the mean should be right around here. That should be the mean. And then the standard deviation of a sampling distribution.

1:26So the standard deviation of this sampling distribution for something with a proportion is going to be equal to the square root of our proportion. So 0.75 times one minus our proportion. So 0.25 divided by the sample size for the sampling distribution. So this is 200.

1:50So the larger the sampling size, the smaller standard deviation we should have for this sampling distribution. So we could get a calculator out to figure out what this is going to be equal to. So if I have 0.75 times 0.25 divided by 200, that gets me to that. And then if I were to take the square root of that, that is going to be equal to 0.03 roughly.

2:24Lemme just write that. So this is approximately equal to 0.03. So if I go from this mean, one standard deviation would get me to... Well, it would actually, you can't go above one, so we have to be careful there. But one standard... Oh, 0.03, I should be very careful.

2:50So 0.75 plus 0.03, 0.78. So one standard deviation above the mean would be right around there. One standard deviation below the mean would be right around there. Two standard deviations would get us right around there. Three standard deviations would get us right around there.

3:13So what I would expect this distribution to look like, and I'm gonna hand-draw it, it should be pretty closely centered around the mean, and maybe look something... Lemme do this in a darker color, and it should actually be a fairly narrow distribution because these are actually fairly large sample sizes.

3:35It should look something like that. I'm not doing it perfectly, but it should look something like that. So something like that. So is this simulation consistent with it? Well, it's not completely off, but I would say the simulation's mean for the sampling distribution looks, just eyeballing it, to be below the theoretical mean of the sampling distribution, and the simulation's standard deviation, the standard deviation of the simulated distribution seems to be larger.

4:09So I would say roughly no. And that doesn't mean that Emanuel's doing anything wrong, it's just by some probability you might just be getting samples that are maybe more skewed than not.