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20 October 2003 5.2 The Sampling Distribution of a Sample Mean The expected mean of x-bar The expected variance of x-bar The expected histogram of x-bar Sampling from a known population Suppose we have a population whose mean and variance (or standard deviation) is known. Suppose we draw a simple random sample (SRS) of n people from that population, and calculate the mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Suppose we draw another SRS of n people from that population, and calculate its mean. Sampling from a known population After drawing lots and lots of simple random samples, we will have a list of lots and lots of sample means. (The mean will vary from sample to sample, depending on just who gets selected.) What should we expect these sample means to look like? The Central Limit Theorem If you draw simple random samples of size n from a population with mean m and variance s2 then the expected mean of x-bar is m the expected variance of x-bar is s2 / n the expected histogram of x-bar is approximately normal If you draw simple random samples of size n from a population with mean m and variance s2 mx m If you draw simple random samples of size n from a population with mean m and variance s2 s 2 X s n 2 If you draw simple random samples of size n from a population with mean m and variance s2 s X s n If you draw simple random samples of size n from a population with mean m and variance s2 The expected histogram of the mean will be approximately normal if n is large, or if the histogram of the population is approximately normal. If you draw small samples from a very non-normal population, then you should not expect your means to have an approximately normal histogram. The expected histogram of the mean will be approximately normal if large sample small sample approx normal population non-normal population The expected histogram of the mean will be approximately normal if large sample small sample approx normal population non-normal population The expected histogram of the mean will be approximately normal if large sample small sample approx normal population non-normal population The expected histogram of the mean will be approximately normal if large sample small sample approx normal population non-normal population The expected histogram of the mean will be approximately normal if large sample small sample approx normal population non-normal population A Central-Limit-Theorem Example Heights of American adult women are approximately normally distributed, with a mean of 64.5 inches and a standard deviation of 2.5 inches. If we draw samples of 100 women from the population over and over again, then we would expect that the means of these samples would have an approximately normal distribution, with a mean of 64.5 inches and a standard deviation of 0.25 inches. According to Table A, about 68% of our samples should have means between 64.25 and 64.75 inches. According to Table A, about 95% of our samples should have means between 64 and 65 inches. Homework Problem Sets Homework 7 27 Oct 5.2 28, 31, 36, 46, 49 Homework 8 10 Nov 6.1 3, 4, 7, 10, 16, 17, 19