 Hi folks, this is Dr. Don. I have a problem out of Chapter 5, Part 4, on the normal distribution and the central limit theorem that a number of folks have struggled with, so I thought I would go over it. It's about a manufacturer that claims the lifespan of its tires is 48,000 miles. This person, you work for a consumer protection agency, you're testing the tires, you're told to assume the lifespan of the tires are normally distributed, that means the population is normally distributed, you take a sample of 100 randomly and you test them, you get a mean lifespan of 47,867, you assume the population standard deviation sigma is 800. Three parts to the question, the first is, assuming the manufacturer's claim is correct, what is probably the mean of the sample is 47,867 or less? Part B is using the answer from Part A, what do you think the manufacturer's claim? The claim that the lifespan is 48,000 miles. Then Part C, assuming the manufacturer's claim is true, would it be unusual to have an individual tire with a lifespan? Now those, I emphasized individual tire here and mean of the sample there, because those are key things you need to pick up on when you're reading this. You're given the clue that the population is normally distributed. From looking at this, given the fact that you're given the mean of the population's standard deviation of the population and a mean of a sample, that should clue you that this is a normal distribution problem. So I want to show you how to solve it using StatCrunch. Always like to start these problems by making a sketch. We're told that the population of tires' milages are normally distributed, so I created the bell curve there, of course. We're given a mean of the population mu of 48,000, and then we're given the mean of the samples of our tires, x bar of 47,867, which is down here on the lower side. We're asked to find out the probability of getting 47,867 or less, which is everything to the left of that value. So we're interested in this area under the curve in red here. Again, we're given the mean of the sample x bar of 47,867, the mean of the population of tires of 48,000, and the standard deviation of the population of tires sigma of 800. Recall when we're dealing with normal distributions and we're trying to find probabilities, we first have to find the z-score. In the z-score, we find from an equation like this, x, the value that we're concerned with, minus the mean mu divided by the standard deviation sigma. Now this equation applies to an individual x when we're looking at the population. When we're looking at a sample, we have to look at it in terms of x bar, the mean of the sample, and we have to subtract from it mu sub x bar, which is the mean of the sampling distribution, and divide that by sigma sub x bar, which is the standard deviation of the sampling distribution, also called the standard deviation of the sample means. That in turn, we can come up with the equation x bar minus mu sets for a normal population, and when we have a sample greater than 30, we know that the sample is normally distributed, and the mean of the sample, mu sub x bar, is equal to mu, the population mean. The standard error, sigma sub x bar, is equal to sigma, the population standard deviation divided by the square root of n, the sample size. So that's how we would manually solve that. We can do it very quickly using StatCrunch. Let's go do that now. Have StatCrunch open here. We go to the path, StatCalculatorNormal to bring up a normal distribution calculator. Recall that it comes up in the standard normal distribution, which has a mean of zero and a standard deviation of one. We're going to work with the population mean of 48,000, and because we're working with the sample first, we need to take the sigma, which was 800, and we're going to divide that by the square root sqrt, open 100 for our sample size, and then we put in our x bar of 47867, and we click compute. It gives us an area under the curve that probably we're looking for of .048, which is approximately .05. If you recall, we had two ways of deciding if something is unusual. If it's two standard deviations or less, then that's not unusual. And if you recall from the empirical rule, two standard deviations means that 95% of the data is within that, so that leaves approximately 5% that would be left over out here. So that kind of tells us, just by looking at the probability, that this is not unusual because it's about 5%, which is not unusual. But let's find the z value just to double check that, and I'm just going to open up another calculator here. This time, the way I like to do it, we've got our standard normal with a 0 and 1, but the mean and standard deviation, I take this probability and I copy it, and I paste it over here, make sure my direction again, we want the left tail, click compute, and that gives us our z value that we were looking for. This minus 1.66, which is less than 2 as we talked about, so that's not unusual. Now let's do the last question, what is the probability of getting a single tire from that population with a mileage life of 47, 8, 67? We can do that using the two calculators I have there. I'm going to make this go back to just the population sigma of 800, click compute again, and you can see now that the area under the curve is 0.333, which is obviously not unusual. And to get the z value, I'm just going to copy that again, ctrl-c, and I'm going to paste it over here, ctrl-v, click compute, and you can see that is minus 0.166z score, obviously much less than two standard deviations, therefore it's definitely not unusual. So that's how to do this quickly using stat crunch, I hope it helps.