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Sampling distribution of sample proportion part 2

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.K (LO)
,
UNC‑3.K.1 (EK)
,
UNC‑3.K.2 (EK)

Video transcript

this right over here is a scratch pad on Khan Academy created by Khan Academy user charlotte allen and 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 when the yellow gumballs we said 60% were yellow so let's make 60% here green and then let's take samples of 10 just like we did before and then let's just start with one sample so we're gonna 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% 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 and so notice now we see here on this distribution two of them were had 50% green and we could keep drawing samples and let's just really increase so we're gonna do 50 samples of 10 at a time and so here we can quickly get to a fairly large number of samples so here were over a thousand samples and what's interesting here is we're seeing experimentally that our sample the the mean of our sample proportion here is 0.6 to 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 and 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 too close to one this looks pretty close to a normal distribution and 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% zero point one what do you think the distribution is going to look like that 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 and if your mean is closer to 0 then you're gonna see the meat of your distribution here and then you're gonna see a long tail to the right which creates that right skew and if your population proportion was close to 1 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 and so let's let's do a population proportion that is right in between and so here this is similar to what we saw before this is looking roughly normal but now and that's when we had 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 1 yet but it is a much tighter distribution and the reason why that made sense the standard deviation of your sample proportion it 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 it's me and it's standard deviation and you feel good about it because we saw it in a simulation
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