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ioe265f11-Lec14

# ioe265f11-Lec14 - Lec 14 Sampling Distribution/Central...

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Lec 14 - Sampling Distribution/Central Limit Theorem IOE 265 F11 1 1 Sampling Distribution Central Limit Theorem 2 Topics I. Concepts of a “Statistic” II. Sampling Distribution of Statistics By Probability Rules Distribution of sample mean and sample total III. Central Limit Theorem IV. Distribution of a Linear Combination

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Lec 14 - Sampling Distribution/Central Limit Theorem IOE 265 F11 2 3 I. Concepts of a “Statistic” Consider taking two samples of size n from the same population distribution. A: 30.7, 29.4, 31.1 Mean 30.4 B: 28.8, 30.0, 31.1 Mean 29.97 Propositions Any sample statistic (e.g. sample mean, sample variance) varies from sample to sample. Therefore a sample statistic behaves like a random variable! 4 Example: Minitab Suppose X ~ Weibull (shape= 2, scale = 5) E(X) = 4.4311; V(X) = 5.365 Using Minitab obtain 6 samples of 10 observations each and calculate sample mean and variance for each. Sample 1 2 3 4 5 6 Mean 4.401 5.928 4.229 4.132 3.620 5.761 Median 4.360 6.144 4.608 3.857 3.221 6.342 Std Dev 2.642 2.062 1.611 2.124 1.678 2.496
Lec 14 - Sampling Distribution/Central Limit Theorem IOE 265 F11 3 5 Point Estimates / Sampling Distributions Point Estimate – value for a sample statistic from a particular sample. Statistic – rv whose value may be calculated from a sample of data -- lowercase letter indicates the calculated or observed value of the statistic. ; S s Probability Distribution of a Statistic is known as its Sampling Distribution. x X value computed the is where ) ( x x X P 6 iid Random Samples Sampling Distribution depends on several items: Population Distribution (parameters) Sample size, n Method of Sampling (with or without replacement) rv’s X 1 , X 2 , .. X n form a random sample of size n if: 1. Xi’s are independent rv’s ( independent ) 2. Every Xi has the same probability distribution ( identically distributed ) If satisfy above two conditions we say Xi’s are iid sampling with replacement or from infinite population iid sampling w/o replacement requires sample sizes n much smaller than population N to assume iid (rule: n/N <= 0.05).

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Lec 14 - Sampling Distribution/Central Limit Theorem IOE 265 F11 4 7 II. Deriving Sampling Distribution of a Statistic By Probability Rules used for simple cases with a few Xi’s cases where derivation is already done. By Simulation (more common!) typically used when derivation via probability rules is complicated, or if: Underlying distribution of interest in unknown (assumed). 8 Deriving Via Simple Probability Example: Suppose you sell two brands of DVD players for A: \$150, and B: \$200. Sales records indicate the following: A – 60% of Sales; B: 40% of Sales Let X 1 – revenue from first DVD; X 2 revenue from second DVD Suppose you take samples of size n=2.
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