06.testing - Lecture 6: Review of Statistics (Last Part)...

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Lecture 6: Review of Statistics (Last Part) Hypothesis Testing and Interval Estimation BUEC 333 Summer 2009 Simon Woodcock
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The Importance of Sampling Distributions Why all the fuss about sampling distributions? Because they are fundamental to hypothesis testing . Remember that our goal is to learn about the population distribution. In practice, we are interested in things like the population mean, population variance, some population conditional mean, etc. We estimate these quantities in a random sample taken from the population. This is done with sample estimators like the sample mean, the sample variance, or some sample conditional mean. Knowing the sampling distribution of our sample estimator (e.g., the sampling distribution of the sample mean), gives us a way to assess whether particular values of population quantities (e.g., the population mean) are likely or unlikely. E.g., suppose we calculate a sample mean of 5. If the true population mean is 6, is 5 a “likely” or “unlikely” occurrence?
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Hypothesis Testing We use hypothesis tests to evaluate claims like: the population mean is 5 the population variance is 16 some population conditional mean is 3 When we know the sampling distribution of a sample statistic, we can evaluate whether its observed value in the sample is “likely” when the above claim is true . We formalize this with two hypotheses. For now, we’ll focus on the case of hypotheses about the population mean, but we can generalize the approach to any population quantity.
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Null and alternative hypotheses Suppose we’re interested in evaluating a specific claim about the population mean. For instance: “the population mean is 5” “the population mean is positive” We call the claim that we want to evaluate the null hypothesis , and denote it H 0 . H 0 : μ = 5 H 0 : μ > 0 We compare the null hypothesis to the alternative hypothesis , which holds when the null is false . We will denote it H 1 . H 1 : μ ≠ 5 (a “two-sided” alternative hypothesis) H 1 : μ ≤ 0 (a “one-sided” alternative hypothesis)
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How tests about the population mean work Step 1: Specify the null and alternative hypotheses. Step 2a: Compute the sample mean and variance Step 2b: Use the estimates to construct a new statistic, called a test statistic , that has a known sampling distribution when the null hypothesis is true (“under the null”) the sampling distribution of the test statistic depends on the sampling distribution of the sample mean and variance Step 3: Evaluate whether the calculated value of the test statistic is “likely” when the null hypothesis is true. We
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This note was uploaded on 06/21/2009 for the course BUEC 333 taught by Professor Simonwoodcock during the Summer '09 term at Simon Fraser.

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06.testing - Lecture 6: Review of Statistics (Last Part)...

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