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Unformatted text preview: M4056 Lecture Notes. September 8, 2010 Sampling from a normal distribution II Before getting to the meat of this lecture, we provide some orientation. Bear in mind the kinds of inferential tasks we might face: i ) We are sampling from a population known to be normal and with known variance, seeking to determine the mean. The previous lecture contained all the information we need make the optimal guess about the mean and to compute confidence intervals. We will examine the practical details later. ii ) We are sampling from a population known to be normal but with unknown variance, seeking to determine the mean. This is a much more common situation than i ). In this case, we can still make an optimal guess about the mean, but we cannot find confidence intervals without information incorporating some estimate of the variance. The present lecture begins to address the mathematical theory that supports this. iii ) We are sampling from a population not known to be normal. The theory we are presenting now has no bearing on this. Throughout todays lecture, we will be dealing with a sample X 1 , . . ., X n of size n drawn from an n ( , 2 ) distribution. X denotes the sample mean and S 2 denotes the sample variance. Last Friday, we saw that X would be n ( , 2 /n ). Today, we will show: Fact 1. X and S 2 are independent. (5.3.1.a) Fact 2. ( n- 1) S 2 / 2 has the same distribution as the sum of the squares of...
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