Confidence Intervals and the Central Limit Theorem

All right, we're going to continue to talk about the normal distribution in this video. And in particular, we're going to learn how to actually calculate the confidence interval from the normal distribution. And so, in the past we have defined our 95 percent confidence interval as the mean, plus or minus the two times the standard deviation. And so, in effect that looked something like using our data from above where we calculated the mean and standard error of our p hat distribution, meanR minus 2 times SER comma meanR plus 2 times SER. Then we can print the confidence interval. And so, here we can see that the 95 percent confidence interval is 0.37 to 0.63. But this is an approximation. Normally our value here, is usually calculated from something known as a z star, z, or a z statistic. So in order to do that, we need to specify the interval, P equals 95. And then we calculate the zstar which is just using stats dot norm dot interval of P divided by 100. And what we want to do is convert this into an array so we can run that. We can look at zstar which is minus 1.95 plus 1.95. And we can see how that is very close to 2. So what we have been doing so far in this class is using two as a very easy approximation for the actual z-star value, which is 1.95996398. So if we want to do a confidence interval with this zstar, we can really use the same formula. I'm going to copy this down. But instead of 2, we can use the actual value. So, and we don't need to do an array because this value already has the negative and the positive here. So I'm going to erase that, still say meanR but replace 2 with zstar times the standard error for R. And so, we can run this. And we can see that it has flipped it until we do this, change this to a plus. It goes back to the order, smaller versus larger because that's adding a negative and then adding a positive. And so we can see that it actually is very similar once it gets out to this third digit is where we start to see differences in the confidence interval. But the benefit to doing this type of methodology is that we can easily change our confidence interval to whatever confidence level we want by just changing this p-value. So if we want the 80th confidence interval, all I need to do is change that to 80. My zstar changes and now my confidence interval has changed. And so, this is a very easy way to do confidence intervals. And it's something that you would normally do in a traditional statistics class where you would look up this zstar value or your dip in confidence level and do a calculation. But it's important to note that this style of doing confidence intervals assumes that your data is normally distributed, and that it is sufficiently large enough to do this type of statistical analysis on it because it is following the central limit theorem, which we talked about earlier in the lesson.