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• Chapter 7, Sample Distribution –A sampling distribution is a distribution of all of the possible values of a statistic (say sample mean) for a given size sample selected from a population. • Sample Distribution of the Mean is an Unbiased Estimate of the Population Mean –If all possible samples of a certain size, n, are selected from a population, the mean of these sample means (the grand mean) would be equal to the population mean. μX μ • Assume there is a population of size N=4 • Random variable, X, is age of individuals • Values of X: 18, 20, 22, 24 (years) • The population mean and standard deviation are: μ σ X i N 18 20 22 24 21 4 (X μ) i N P(x) 2 2.236 18 A 20 B 22 C 24 D Uniform Distribution Now consider all possible samples of size n=2 drawn from this population 1st Obs 2nd Observation 18 20 22 24 1st 2nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 18 18 19 20 21 20 20,18 20,20 20,22 20,24 20 19 20 21 22 22 22,18 22,20 22,22 22,24 22 20 21 22 23 24 24,18 24,20 24,22 24,24 24 21 22 23 24 16 possible samples (sampling with replacement) 16 Sample Means • Sampling Distribution of All Sample Means P(X) Notice: 18 19 20 21 22 23 24 (no longer uniform) μX X i N σX ( X 18 19 21 24 21 16 i μ X )2 N (18 - 21) 2 (19 - 21) 2 (24 - 21) 2 1.58 16 Population N = 4, μ 21 Sample Means Distribution n = 2, μ X 21 σ 2.236 σ X 1.58 • Why std. Dev. Of the means distribution is smaller than that of the population? Reasons: • Different samples of the same size from the same population will yield different sample means • A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: σ σX n •Note that the standard error of the mean decreases and the distribution becomes less dispersed as the sample size increases (see page 235) • If a population is normally distributed with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with μX μ σX σ n • Z-value for the sampling distribution of Z where: ( X μX ) σX ( X μ) σ n X = sample mean σ = population mean μ n X is calculated: = population standard deviation = sample size • If population is not normally distributed, we can apply the Central Limit Theorem which proves that: – …sample means from the population will be approximately normal as long as the sample size is large enough. • What is large enough? • For most distributions, n > 30 will give a sampling distribution that is nearly normal • For fairly symmetric distributions, n > 15 • For normal population distributions, the sampling distribution of the mean is always normally distributed • (See page 238 for distributions of the population and samples) Application: A brand name breakfast cereal company produces 5000 boxes of serial per day. Each box is suppose to have 368 grams of cereal with an average dispersion of 15 grams. • Set up the information in terms of population distribution. • Questions: 1. What percent of individual boxes will have less than 365 grams? 2. If a sample of 25 boxes are selected what is the probability that the sample mean is less than 365? 3.If all possible samples of size 25 are taken, what interval around the population mean will contain 95% of all sample means? 4.What is the probability that a sample mean will be within the above estimated interval? • All material in this chapter assumed sampling with replacement. Apply the Finite Population Correction (fpc) if: – the sample is large relative to the population (n is greater than 5% of N) and… – Sampling is without replacement N n • The fpc factor is N 1

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