 Suppose you know that a job will take two hours to complete. You could tell the customer that the repair will be completed in two hours, but no one ever complains when a job is completed ahead of schedule. So you'd probably tell the customer the repair will take six hours. And so if you took three hours, the customer who was told two hours will be irate, but the customer who was told six hours will be ecstatic. And now you know how to gain the reputation as a miracle worker. The central limit theorem gives us a way to produce a confidence interval in terms of the population's standard deviation. And we saw earlier that we have three common confidence intervals, 90% is plus or minus 1.64 standard deviations, 95% is plus or minus 1.96 standard deviations, and 99% is plus or minus 2.58 standard deviations. There's just one problem. We don't know the population's standard deviation. So in practice, we use the sample standard deviation. How does that work? Well, let's go back to our light bulbs. So here we find a sample of 100 light bulbs to have a mean lifespan of 9,800 hours and a standard deviation of 100 hours. We want to find a 95% confidence interval for a population mean and interpret the interval. The sample mean will be normally distributed, except we don't know the population mean or the population standard deviation. Actually, we don't use the population mean, so it doesn't matter that we don't know what it is. We do need the population standard deviation, so we'll assume that it's equal to our sample standard deviation. So we can compute the standard deviation of the mean. And since we're looking for a 95% confidence interval, our confidence interval will be within 1.96 standard deviations of the sample mean. So they'll go from sample mean minus 1.96 standard deviations, up to the sample mean plus 1.96 standard deviations. And so this interval will be, and importantly, this means that if the mean falls in this interval, the sample mean we observed is part of an event that will occur 95% of the time. Again, it's very important to remember that the 95% confidence interval does not indicate the population mean has a 95% probability of being in the interval. Or for example, suppose we have a sample of 400 gizmos with a mean mass of 85 grams with a standard deviation of 0.8 grams. Let's find an 80% confidence interval for the population mean. We'll assume our population standard deviation is our sample standard deviation and find the standard deviation of the sample means, which will be. Now since we don't know the width of the 80% confidence interval, we should see how wide the interval needs to be to give us an 80% probability. So the probability that we're within one standard deviation of the mean is 68%, which is not 80%. So let's expand our interval if we go up to 1.64 standard deviations of the mean. The probability we're within 1.64 standard deviations is 90%, which is a little too much. So let's narrow our interval. So the probability we're within 1.32 standard deviations is a little bit over 81%. We're almost there, so let's narrow it just a little more. And at 1.28 standard deviations, we have our 80%. So our 80% confidence interval should be plus or minus 1.28 standard deviations of the mean. And so the 80% confidence interval will be plus or minus 1.28 standard deviations from the sample mean. And so we find.