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# Class Note for STAT 528 at OSU 51

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Date Created: 02/06/15

Stat 528 Autumn 2008 Elly Kaizar The sampling distribution for the sample mean Reading Section 52 o The mean of the sampling distribution 0 The variance and stdev of the sampling distribution 0 Normal populations 0 Non normal populations 0 Remarks The sampling distribution for the sample mean 0 Suppose we collect a SR8 of size n c We collect measure a single observation on each individual in the sample Xi i 1 n o The sample mean is Y 2211 Xi H o What is the sampling distribution of Y o ls there a good approximation to the sampling distribution The sampling distribution of the sample mean 0 Let X1 X2 Xn be a SR8 of size n drawn from a popu lation with mean u and standard deviation 0 0 Then the mean of the sample mean is My M o The variance of the sample mean is 2 02 07 X n o The mean value of the sample mean is the population mean the sample mean is an unbiased estimator of the popula tion mean77 it estimates What it is supposed to o The variance and standard deviation of the sample mean de crease as the sample size 71 increases Sampling distribution of the sample mean the normal population case 0 Let X1 X2 Xn be a sample of size n drawn randomly from a normal distributed population with parameters 1 and 0 Then 2 Y has a Nltu 0 distribution 71 o The sample mean is normally distributed 0 Why 0 A useful result a linear combination of indepen dent normal RVs is normal Suppose X and Y are independent normals RVs and a and b are constants Then aX bY is also a normal RV Sampling distribution of the sample mean the nonnormal population case 0 Let X1 X2 Xn be a sample of size n drawn randomly from a population with mean u and standard deviation 0 We assume the mean and variance are nite numbers 0 The Central Limit Theorem A good approxima tion when n is large enough is that the sampling distribu tion of the sample mean is normal with mean u and variance 0271 Alternatively we write 2 a X has approximately a N u gt distribution 71 for large enough 71 The nonnormal case some remarks o The great thing about the Central Limit Theorem We do not need to know how the population is distributed to know or at least approximate the distribution of the sam ple mean For any given population and targeted quality of approximation there is a large enough 71 so that the distri bution of the sample mean is approximately normal 0 The more discrete the population distribution the larger 71 we need for the approximation to hold 0 The less symmetric the population distribution the larger 71 we need for the approximation to hold 0 As well as n we do need to know u and a or be able to guess estimate them Piston ring example The inside diameter of a randomly selected piston ring is a RV With a mean of 12cm and a standard deviation 004cm 0 Where is the sampling distribution of Y centered What is the stdev of the sampling distribution of Y o What are the mean and stdev based on a random sample of size 64 Of size 16 0 Which is more likely to lie Within 001cm of 12cm the mean of a random sample of size 16 or the mean of a random sample of size 64 Why Chicks example An experiment to compare the nutritive value of normal corn and high lysine corn divides 4O chicks at random into two groups of 20 One group is fed a diet based on normal corn while the other receives a high lysine corn At the end of the experiment inference about which diet is superior is based on the difference 7 7 between the mean weight gain 7 of the 20 chicks in the high lysine group and the mean weight gain 7 of the 20 in the normal corn group We assume that the two sample means are independent Chicks example cont 0 Suppose that uX 360g and 0X 55g in the population of the Chicks fed normal corn and that My 385g and 05 50g in the high lysine population What is mean and stdev of 7 Y o The weight gains are normally distributed in both popula tions What is the distribution of Y Of 7 What is the distribution of the sampling distribution of 7 Y o What is the probability that the mean weight gain in the high lysine group exceeds the mean weight gain in the normal corn group by 25g or more

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