part6 - Public Health 6450 – Fall 2011 Andrew Mugglin and...

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Unformatted text preview: Public Health 6450 – Fall 2011 Andrew Mugglin and Lynn Eberly Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 06 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Where are we going? Previously: • Probability! • More on random variables Current topics: • Review of some important concepts • Binomial distribution: definition • Sampling distribution of the Binomial mean Mugglin and Eberly PubH 6450 Fall 2011 Part 06 2 / 45 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Turning Point TP6a: Bayes Theorem Mugglin and Eberly PubH 6450 Fall 2011 Part 06 3 / 45 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Statistics and sampling distributions (revisited) In Part 4, we saw the ideas of statistics, sampling distributions, and bias: Definitions The population distribution of a random variable is the probability distribution of its values for all members of the population. A statistic (e.g., mean, count, proportion) is also a random variable. The probability distribution of the statistic is its sampling distribution . A statistic is unbiased if it neither consistently underestimates nor overestimates the value of the corresponding population parameter. For example, the mean ¯ x from a simple random sample is unbiased for the population mean μ . Mugglin and Eberly PubH 6450 Fall 2011 Part 06 4 / 45 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Mean and variance of an average (revisited) In Part 5, we learned about computing expectation and variance: • E[X] = population mean • E[aX + aY] = aE[X] + aE[Y] • Var[X] = population variance • Var[aX] = a 2 Var[X] • If X and Y are independent, then Var[X + Y] = Var[X] + Var[Y]. We used these rules to show that E [ Z ] = E [( X + Y ) / 2] = μ and Var[ Z ] = σ 2 / 2. Mugglin and Eberly PubH 6450 Fall 2011 Part 06 5 / 45 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Mean and variance of an average (cont’d) We asked you to go home and show that E [ ¯ X ] = μ and Var[ ¯ X ] = σ 2 / n for a sample of size n. Answer: E ¯ X = E 1 n ( X 1 + X 2 + ··· + X n ) = 1 n E [ X 1 + X 2 + ··· + X n ] = 1 n ( E [ X 1 ] + E [ X 2 ] + ··· + E [ X n ]) = 1 n ( μ + μ + ··· + μ ) = 1 n ( n μ ) = μ Mugglin and Eberly PubH 6450 Fall 2011 Part 06 6 / 45 Review Review of important concepts Defining the Binomial Distribution Sampling Distribution of the Binomial Mean Mean and variance of an average (cont’d) Var ¯ X = Var 1 n ( X 1 + X 2 + ··· + X n ) = 1 n 2 Var [ X 1 + X 2 + ··· + X n ] = 1 n 2 (Var[ X 1 ] + Var[ X 2 ] + ··· + Var[ X n ]) = 1 n 2 ( σ 2 + σ 2 + ··· + σ 2 ) = 1 n 2 ( n σ 2 ) = σ 2 / n Mugglin and Eberly PubH 6450 Fall 2011 Part 06 7 / 45 Review...
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This note was uploaded on 11/21/2011 for the course PUBH 6450 taught by Professor Andymugglin during the Fall '10 term at Minnesota.

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part6 - Public Health 6450 – Fall 2011 Andrew Mugglin and...

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