 This is a video on how to use the Central Limit Theorem, which uses ideas regarding the normal distribution. The population of weights for men attending a local health club is normally distributed with a mean of 182 pounds and a standard deviation of 26 pounds. An elevator in the health club is limited to 34 occupants, but it will be overloaded if the total weight is an excess of 6,664 pounds, which is dangerous when an elevator is overloaded. Assume there are 34 men in the elevator, what is the average weight which the elevator would be considered overloaded? So average weight, that means I have to take the 6,664 pounds and divide it amongst the 34 people. If the average weight was 196 pounds among those 34 people, then the elevator would be overloaded. Now I want to know what is the probability that one randomly selected male or one data value will exceed this weight. So the probability that a male, which will represent by x, will be more than 196 pounds. Remember we use x to represent data values. So in my distribution, my mean is going to be 182. I plot 196 over to the right and those that are more than it are those to the right of 196. So we find the area under the curve to the right of 196. Keep in mind that the mean is 182 and the standard deviation is 26. This is what we'll use in Google Sheets. So the mean is 182, standard deviation is 26. Lower bound is 196, upper bound is infinity or six nines. You can do more nines if you want, but it really doesn't matter. And you get about 0.2951. That would be 0.2951. That is the probability that one male will exceed this weight. But one person exceeding that weight may or may not affect the fact that the elevator is overloaded. It's if all of these people, all 34 of them exceed the average weight that creates an issue. So that's where the central limit theorem comes into place. So assume that 34 male occupants were in the elevator and were randomly selected to find the probability that the elevator will be overloaded. So what the central limit theorem says is that out of the population of men, if I randomly select 34 and I find their average, I'm going to get some sort of value. Then I take another 34 men randomly selected, take their average, find their mean, X bar. Take another 34 out of the population and then take their average. Another 34. Take their average. Another 34. Take their average. The averages of these averages, the mean of these sample means, follows a normal distribution. So I call this my sampling distribution. It's a distribution that talks about the sample means. It's all of my collections of 34 males. I'm interested the probability that these collections of 34 males will be greater than 196. So some things you have to know. The mean of my sampling distribution or the mean of my sample means is the same as the mean of my original information. So that would be 182. What's different, however, is the standard deviation of the sampling means. The standard deviation of the sampling means is the original standard deviation divided by the square root event. Also known as the finite population correction factor. Now my original standard deviation was 26 and in my sample size is 3426 divided by square root of 34. I'm going to go ahead and say that is 4.45896. The more decimal places you keep, the more accurate your answer will be. For my sampling distribution, I have a mean of 182. And I want to know the probability that the means, the mean of the sample means will be more than 196. So area to the right of 196. Using Google Sheets, your mean is still 182. Your standard deviation though has now been adjusted to 4.45896 thanks to the central limit theorem. Your lower bound still the same and your upper bound still the same. You get .0008. You can keep more decimal places if you want, but 0.0008 will be what we use. So this is the probability that the elevator will be overloaded. Once again, I had to use the central limit theorem because we weren't looking at just one male occupant of the elevator. We were looking at a sample of 34 that their average weight would be more than 196. I had to use my sampling distribution, which had the same mean as we previously used, but the standard deviation does have to be altered slightly. So if the elevator is full on average nine times per day, how many times will the elevator be overloaded in one leap year or one non leap year? So that would mean 365 days in a year. So if you have 365 days in a year and the elevator is full nine times per day, that means in a year, the elevator will be full 3,285 times. And of those 3,285 times, it will be overloaded 3,285 times the probability found in the previous section. It's not very often, but hey, let's see how many times per year the elevator is predicted to be overloaded. And you do the multiplication and you get approximately three times per year. Approximately three times per year, the elevator will be overloaded. Is this reason for concern? Yes, we don't want that elevator to be overloaded at all during a year's time. We want our number as close to zero as possible. Three times per year is a reason to be concerned because if the elevator malfunctions three times a year, I'm pretty sure that business or wherever that elevator is, is not going to be staying in business. So yes, there is reason to be concerned here. Thank you for watching.