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Unformatted text preview: Introduction to hypothesis tests 1 Introduction 1.1 Definitions and examples Definition: hypothesis a claim about a statistical model. Examples Population model: Suppose that heights of US residents are modeled with a distri- bution with unknown mean and unknown standard deviation . Claims that = 65 inches, 6 = 65 inches, > 65 inches, < 65 inches, = 4 inches, 6 = 4 inches are all hypotheses about the model for heights of US residents. In addition, claims that heights of US residents are Normally distributed, heights of US resi- dents are not Normally distributed are hypotheses about the model for heights of US residents. Process model: Suppose that each television produced by a factory is independently defective with unknown probability . Let X Bern( ), correspond to whether or not the next television to be produced is defective. A claim that = 5% is a hypothesis about this Bernoulli model. Definition: null hypothesis denoted H , is the hypothesis/model we currently use. Sometimes called the null model. (e.g. the probability of heads for this coin is 50%, which is denoted H : = 0 . 5.) Definition: alternative hypothesis denoted H a , is a hypothesis contradictory to H . Sometimes called the alternative model. (e.g. the probability of heads for this coin is greater than 50%, which is denoted H a : > . 5.) Definition: hypothesis test A method to decide whether or not to reject the null hypothesis (null model) H in favor of an alternative hypothesis (alternative model) H a using data. Example 1.1: We wish to determine if a coin is fair or biased in favor of heads. Let denote the probability of heads for this coin. The null and alternative hypotheses are: H : = 0 . 5 H a : > . 5 The coin was flipped 1000 times and 525 heads were observed, hence the observed sample proportion is p = 0 . 525. Do these results present sufficient statistical evidence to reject H in favor of H a ? 1 Lets assume that H is true, that is, lets assume that = 0 . 5. From corollary A and B, the sample proportion (random variable) P is approximately N ( P , P ), where P = = 0 . 5 and P = r (1- ) n = r . 5 * . 5 1000 = 0 . 01581139 If we planned to independently repeat this experiment of flipping the coin 1000 times, as- suming = 0 . 5, the probability of observing a new sample proportion greater than or equal to our current realization p = 0 . 525 is P ( P . 525), which can be computed approximately with, P ( P . 525) = P- P P . 525- P P ! P Z . 525- . 5 . 01581139 = 1- pnorm (1 . 581139) = 0 . 05692313 Hence, if the coin is actually fair (i.e. if H is true), there is approximately a 5.7% chance of observing a sample proportion as large or larger than our present realization p = 0 . 525....
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- Spring '08