Stats 309 9-1

# Stats 309 9-1 - 9.1 Sampling Distribution of the Mean...

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9.1 Sampling Distribution of the Mean Remember that a parameter is a measurement about a population and a statistic is a measurement about a sample. In class, we can assume population parameters are known or are calculable. However, in real life, it’s not practical to calculate population parameters most of the time. For example, to determine the mean annual income of North American blue collar workers it would be difficult to poll EVERY blue collar worker in North America, which is not possible. However, if we are willing to accept less than 100% accuracy, we can take a sample and use statistical inference to obtain an estimate. Because the results of random samples and experiments include an element of chance, we can’t guarantee that our inferences are correct. What we can do is guarantee that the methods we use usually give correct answers. So, we take a sample and calculate the sample mean. Probably won’t be equal to the population mean, but if we did things correctly, it should be quite close . For the purpose of statistical inference, we need to be able to measure how close is the sample mean to the population mean. How can we do this if we cannot get the population mean? This is what a sampling distribution helps us do.

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We can describe the behavior of a sample statistic by a probability model that answers the question “What would happen if we did this many times?” Our sampling distribution will help us answer this question. One of the tools that will help is the Law of Large Numbers.

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The law of large numbers is the foundation of such business enterprises as gambling casinos. How do casinos make money? The winnings or losses of a gambler on a few plays are uncertain-that’s why gambling is exciting. It is only in the long run that the mean outcome is predictable. The house plays tens of thousands of times. The average winnings of the house on tens of thousands of plays will be very close to the mean of the distribution of winnings, which is in the house’s favor, of course. This guarantees the house a profit in the long run. The law of large numbers assures us that if we measure enough subjects, x-bar
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## Stats 309 9-1 - 9.1 Sampling Distribution of the Mean...

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