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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