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Chapter 9

STAT 1100 Chapter 9: Chpt. 9 - Sampling Distributions

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STAT 1100
Sarah Quesen

STAT & PROBLTY FOR BUS MGT 4/22/17 Notes ❖ Chapter 9: Sampling Distributions ➢ Section 1 – Sampling Distribution of the Mean • Sampling distribution: created by sampling o 1. Draw samples of the same size from a population, calculate the statistic of interest, and then use descriptive techniques to learn more about the sampling distribution o 2. Method relies on the rules of probability and the laws of expected value and variance to derive the sampling distribution ▪ Sampling Distribution of the Mean of Two Dice • The population is created by throwing a fair die infinitely many times • Population mean: μ = ∑xP(x) • Population variance: σ =∑(x-μ) P(x) 2 o Drawing samples of size 2 from the population o x = new random variable created by sampling ̅ • Mean of the sampling distribution of x: μ = ∑x̅(x) ̅ ̅ ̅ • Variance of the sampling distribution of x: σ =∑(x-̅ ) P̅x) ̅ ̅ 2 ̅ o The variance of the sampling distribution of x is exactly half of the variance of the ̅ population of the toss of a die • Standard deviation of the sampling distribution: σ = √σ 2̅ 2x̅ • The distribution of x is different from the distribution of X ̅ o However, the two random variables are related; their means are the same (μ =μ = 3.5) ̅ and their variances are related (σ = σ /2x̅ 2 • Foreachvalueof n,themeanofthesamplingdistributionof xisthemeanofthepopulation ̅ from which we are sampling o (μ ̅μ) • The variance of the sampling distribution of the sample mean is the variance of the populati2n divi2ed by the sample size o (σ =̅σ /n) • The standard deviation of the sampling distribution is called the standard error the mean o (σ = σ/√n) ̅ • The sampling distribution of x become̅ narrower (or more concentrated about the mean) as n increases o Another thing that happens as n gets larger is that the sampling distribution of x ̅ becomes increasingly bell shaped o This phenomenon is summarized in the central limit theorem • Central Limit Theorem: The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of x will resemble ̅ normal distribution o The accuracy of the approximation alluded to in the central limit theorem depends on
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