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notes515fall10chap6 - STAT 515 - Chapter 6: Sampling...

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STAT 515 -- Chapter 6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean μ ) – it’s typically unknown . Statistic = a number that characterizes a sample (example: sample mean X _ ) – we can calculate it from our sample data. We use the sample mean X _ to estimate the population mean μ . Suppose we take a sample and calculate X _ . Will X _ equal μ ? Will X _ be close to μ ? Suppose we take another sample and get another X _ . Will it be same as first X _ ? Will it be close to first X _ ? • What if we took many repeated samples (of the same size) from the same population, and each time, calculated the sample mean? What would that set of X _ values look like? The sampling distribution of a statistic is the distribution of values of the statistic in all possible samples (of the same size) from the same population.
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Consider the sampling distribution of the sample mean X _ when we take samples of size n from a population with mean μ and variance σ . 2 Picture: The sampling distribution of X _ has mean μ and standard deviation n / σ . Notation: Point Estimator : A statistic which is a single number meant to estimate a parameter. It would be nice if the average value of the estimator (over repeated sampling) equaled the target parameter. An estimator is called unbiased if the mean of its sampling distribution is equal to the parameter being estimated.
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Examples: Another nice property of an estimator: we want the spread of its sampling distribution to be as small as possible. The standard deviation of a statistic’s sampling
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notes515fall10chap6 - STAT 515 - Chapter 6: Sampling...

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