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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 μ and unknown standard deviation σ . Claims that “ μ = 65 inches”, “ μ 6 = 65 inches”’, “ μ > 65 inches”, “ μ < 65 inches”, ‘ σ = 4 inches”, “ σ 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 θ . Let X ∼ Bern( θ ), correspond to whether or not the next television to be produced is defective. A claim that θ = 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 : θ = 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 : θ > . 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 θ denote the probability of heads for this coin. The null and alternative hypotheses are: H : θ = 0 . 5 H a : θ > . 5 The coin was flipped 1000 times and 525 heads were observed, hence the observed sample proportion is ˆ p = 0 . 525. Do these results present sufficient statistical evidence to reject H in favor of H a ? 1 Let’s assume that H is true, that is, let’s assume that θ = 0 . 5. From corollary A and B, the sample proportion (random variable) ˆ P is approximately N ( μ ˆ P ,σ ˆ P ), where μ ˆ P = θ = 0 . 5 and σ ˆ P = r θ (1 θ ) n = r . 5 * . 5 1000 = 0 . 01581139 If we planned to independently repeat this experiment of flipping the coin 1000 times, as suming θ = 0 . 5, the probability of observing a new sample proportion greater than or equal to our current realization ˆ p = 0 . 525 is P ( ˆ P ≥ . 525), which can be computed approximately with, P ( ˆ P ≥ . 525) = ˆ P μ ˆ P σ ˆ P ≥ . 525 μ ˆ P σ ˆ 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 ˆ p = 0 . 525....
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This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
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