Populations, Parameters, Statistics, and Sampling

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• E[S 2 ] ≠ σ 2 Bias in the Sample Variance – proof on p. 216 E[S2 ] = σ2 - σ 2 M • this is because the mean used to calculate S will in general be different than the population mean – the sample mean is always exactly centered on the particular sample and so deviations from it underestimate true population deviations 2 2 2 2 2 σ ⎛ N −1 ⎞ 2 ES [ ] = σ − σM= σ − = ⎜ ⎟σ N ⎝ N ⎠ – the sample variance is too small by a factor of (N-1)/N • S 2 is the uncorrected variance and s 2 is corrected – the corrected standard deviation is still slightly biased (only variance is unbiased) » the amount of remaining bias is small though, so it’s typical to use s to estimate σ • unbiased estimate of standard error of the mean ˆ σ M = ˆ σ = N s = N S N −1
Other Issues with Estimators • Pooling two or more samples results in better estimation – frequency weighted average (pp. 219-220) • For small populations, can’t assume sampling with replacement, and so the estimates need to be slightly larger (p.220) • Because the sample mean is typically different than the population mean, it’s useful to define a confidence interval for the population mean – using Tchebycheff inequality, the sample mean is between x ± kσ M with at least probability 1 – (1/k 2 ) • this is not the probability that x is μ – the probability that the interval from this sample covers μ • this is a very rough estimate and additional assumptions regarding the sampling distribution tighten the estimate – for unimodal symmetric sampling distribution, 95% interval with k=3
Page 1 and 2: Populations, Parameters, Statistics
Page 3 and 4: Maximum Likelihood (MLE) • In cal
Page 5: Sampling Distribution of the Mean

Populations, Parameters, Statistics, and Sampling

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