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Random Sampling

Random Sampling. Population Random sample: Independent variables Same probability distribution Statistics Point estimate. Examples. Mean of a single population Variance of a single population Difference in means of two population. Unbiasedness.

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Random Sampling

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  1. Random Sampling • Population • Random sample: • Independent variables • Same probability distribution • Statistics • Point estimate

  2. Examples • Mean of a single population • Variance of a single population • Difference in means of two population

  3. Unbiasedness • An estimator  of some unknown quantity  is said to be unbiased if the procedure that yields  has the property that, were it used repeatedly the long-term average of these estimates would be . • That is to say, E() = .

  4. Example 1 • Given X = (5.8, 4.4, 8.7, 7.6, 3.1) • The Sample Mean • The Sample Median • The 20% trimmed mean

  5. Example 2 • Sample Variance • Expectation of Sample Variance

  6. Variance of a Point Estimator • MVUE – Smallest variance estimator • Theorem If X1,…Xn is a random sample of size n form a normal distribution with mean and variance , then the sample mean is the MVUE for

  7. Example 1 (continued)

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