 This is Dr. Don. I want to go over some problems in Chapter 5, Section 4 and 5 that a number of people have missed. The first one here, 5.4.1, is one of the basic questions. And we're told that we have a population with a mean mu equals 79 and standard deviation sigma equal 8. And they want us to find the mean and standard deviation of a sampling distribution of sample mean. So our statistic of interest is the sample means and what you should come away from Section 4 is that when you have a repeated sample from a population with a given mean, then the mean of those samples, the mean of the sampling distribution is going to be the same as the mean of the population. So it will be 79. The standard deviation of the sampling distribution is equal to the standard deviation of the population sigma divided by the square root of the sample size. This is one of those problems that is so simple that the folks at Pearson Statcrunch haven't really developed a specific tool. So I like to do them using Excel and as you know I like to build workbooks where I can save my practice problems, my homework problems for later use on exams. In here I just made a screenshot and built a simple little Excel calculator over here where I put in my mean, sigma, and n and it will give me my mu sub x bar, the mean of the sampling distribution of means, and sigma sub x bar, the standard deviation of the sampling distribution of sample means. And that is a 79 and a 1. The solution to this next problem depends on your recognizing that we can apply this central limit theorem here. Our sample is 65, which is greater than 30, and that means that the central limit theorem applies. And if you recall, when we can use the central limit theorem we don't really have to worry about what type of distribution the population is. We know that the distribution of the sample means will be approximately normal and therefore we can solve this using the normal distribution. And what we have to do if you use the tables or Excel is to convert these values of the mean, mu of 64.3 and our x value of 65 to z scores. But using Statcrunch we can just knock it out real quickly. Let's bring up Statcrunch. Okay I have Statcrunch open. As usual we start with our stat. This time we go to calculators and bring up the normal distribution calculator. We know that our mean of the population is 64.3 inches. We know the standard deviation of the population is 2.56, but we've got to convert that into the standard deviation of the sampling distribution by dividing by the square root sqrt of our sample size and our sample was 65. So I just put sqrt inside parentheses. We want to know what is the probability that the mean height of the sample is greater than 65. Mean height of the sample and we want greater than which would be the right tail and we get our answer 0.137 which would be the probability of having a sample mean greater than 65 inches. This next question is one that a lot of folks struggle with and there are some ways that you can make this much easier and much less likely to have a dumb dumb by using Statcrunch. So let's read it here. We're given the information that a manufacturer of tires says the lifespan of its tires is 48,000. That would be the mean mileage of the population. You work for an agency, you're testing these tires, you're assuming that the life spans are normally distributed, but you don't really have to worry about that because you're selecting 100 tires which is greater than 30 and that means that the central limit theorem applies and the distribution of the sample means will be approximately normal and we can use the normal distribution to model this. The mean life of the sample is 47,743, sigma, the population standard deviation is 900. The first question, assuming the manufacturer's claim is correct, what is the probability that the mean of the sample is 47,743 or less? We're going to do that using Statcrunch. I have Statcrunch and again we go to the stat tab, calculators normal, bring up the normal distribution calculator. We know that the mean of the population is 48,000. We know that the standard deviation of the population is 900 and because we're working with a sample this time we have to divide that by the square root sqrt of n and our sample size 100. Put that in parentheses. We want to know what is the probability it'll be 47,743 or less so we want the arrow pointing that way. We click compute and we see that the probability is .002. It's this little bitty red arrow down there, it's very small. We saw that the probability of the sample mean being 47,743 or less is .002, very small number, 2 tenths of 1% chance that it would be that small if the mean of the population is really 48,000 miles. So part B says, what do you think of the manufacturer's claim? The correct answer, the claim is inaccurate because the sample mean would be considered unusual since it does not lie within a range of an usual event, namely within two standard deviations of the mean of the sample means. I'll show you how to calculate how many standard deviations this is, but something you should learn and you'll use going forward in the rest of the course is that if the probability of an event is 5% or less it's unusual. The way that ties in to this question, if you go back to the empirical rule, back in our descriptive statistics, we said that there is a 95% probability that the data will be within two standard deviations of the mean in a normal distribution. So if we've got 95% chance that it's within two standard deviations, the fact that this is only 2 tenths of 1% that is much smaller than the 5% that's left over from the 95% that means it's unusual. So you can use that as a quick way to answer these questions, but let me show you how to get the standard deviations. I'll show you that on part C. Part C says, assuming the manufacturer's claim is true, would it be unusual to have an individual tire, not a sample, an individual tire with a lifespan of 47,743? Why or why not? So let's get the standard deviation. I'm back with the StatCrunch Normal Calculator we used on part B and because this is an invisible tire, we use the population standard deviation, so I'm just going to highlight that square root of 100, square root of n, and delete that. We still have the same information, the mean's 48,000 and we're interested in the probability of an individual tire having a life of 47,743 or less. Click compute and now we've got a large wet area there which is 38%. Using the rule of thumb that I just told you, the 5%, this is much, much greater than 5%, so it's much more likely to happen, so it would not be unusual to have an individual tire of 47,743 if the mean of the population is 48,000 standard deviation of 900. But let's get the Z values. The way I like to do that is I highlight that and use control C to copy, control Charlie, and then I'm going to convert this back to the standard normal which is a mean of 0 and a standard deviation of 1 and we're going to put our cursor inside the probability block and use control V, control Victor to place that probability back there and then click compute. We have now the Z value for that individual tire and it is minus 0.2, minus 0.3 which is much less than two standard deviations. That tells us again that it is not unusual. So that's a quick way to go back and forth using the stat crunch normal calculator between probabilities and Z values. So I hope this helps.