Comparing simulated and theoretical sampling distributions - means

0:00- [Instructor] We're told a population has mean of 80 and a standard deviation of 15. Shannon simulates a large number of samples of size n=100 taken from this population. The distribution of means from her simulation is shown. So what's happening here is each of these dots represent a sample of 100 that Shannon took from this population that had these parameters, mean of 80, standard deviation of 15.

0:25And for each sample, she calculated that sample mean. And so, she, based on that, each of... And then she plotted it. And so this dot right over here shows one of several of the times that she got a sample mean of a little bit under 82. And she did this a ton of time to get this simulated sampling distribution.

0:48Now they ask us, "Is Shannon's simulation consistent with the theoretical sampling distribution of this population for samples of size 100?" Pause this video and have a go at that. All right, so when we're thinking about a theoretical sampling distribution, the mean of a theoretical sampling distribution should theoretically be equal to the mean of the population.

1:10It should be equal to that population parameter. And the standard deviation of the sampling distribution of the means should be equal to the population's standard deviation divided by the square root of the sample size. Now in this case, the mean of our population, we know, is 80.

1:34They gave us that right over there. The standard deviation of our population is 15, and then the sample size is 100. So this one right over here is gonna be 15 divided by 10, which is equal to 1.5. So what we would expect to see for our theoretical sampling distribution of the means, theoretically, is a mean right over here at 80.

1:58I'll draw a little line there. And then let's see, one standard deviation for the theoretical sampling distribution of the means, it should be 1.5 to the right for one standard deviation above the mean. It should be three for two standard deviations, and then it should be at 78 1/2 for one standard deviation below the mean and then at 77 for two standard deviations below the mean.

2:26And let's see, based on that, if I can sketch what this distribution should look like. So I'm just eyeballing it, but it should, in theory... Actually, it should be a little bit more like... In theory, I'm gonna try to fit it as well as I can. In theory, it should look something like that.

2:47And the way my brain is thinking about it is in a normal distribution, by the time you get to one standard deviation in height, you're usually a little bit more than a quarter of the way down, maybe even a third of the way down. By two standard deviations, you're a little bit more of half of the way down.

3:08That's the way I'm actually just trying to ballpark-draw this normal distribution with these properties. And I would say that this simulated sampling distribution actually looks pretty darn consistent with the theoretical sampling distribution that we tried to draw.

3:26So, I would say yes.