lect2 - A Quick Review of Hypothesis Testing In this...

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Unformatted text preview: A Quick Review of Hypothesis Testing In this lecture we will quickly review the following • The basic one sample T test as an example • The decision procedure • Type I and Type II error • OC curves and sample size selection • Practical vs. statistical significance • The relationship between confidence intervals and hypothesis tests A Hypothesis Test has 2 basic components: 1. Hypotheses: Null Hypothesis H Alt. Hypothesis H 1 (essentially "not H ") e.g. H : μ =10 H 1 : μ ≠ 10 2. Decision Criteria It works like this Sample data --> Criteria --> reject or fail to reject H There are four possible outcomes from a Hyp. test: 1. Fail to reject H when H is true (we made the correct decision) 2. Reject H when H is indeed false (right again) 3. Reject H when H is true (a Type I error) 4. Fail to reject H when H is false (a Type II error) Define: α = Pr{we make a Type I error} = Pr{reject H | H true} β = Pr{we make a Type II error} = Pr{fail to reject H | H is false} How good a test is is determined by these probabilities. Example: Recall the one sample T-test Assumptions: 1. Population is NID ( μ , σ 2 ), 2. μ and σ are unknown population parameters H : μ = μ 0 where μ is some specified constant H 1 : μ ≠ μ Decision criteria: Compute the T statistic from the data: n S X T μ- = If |T | > T c then reject H 0 (T c is a “critical” value from a table) Why does this work? 1. The test statistic "measures something significant" about H . If the sample average is far from the hypothesized value, H is likely to be false. 2. We know the distribution of T when H is true. In this example, the test statistic T will be close to zero when H is true. Conversely, T will be far from zero when H is false. Thus it measures something significant about H . If T is far enough away from zero, we can reject H . But how far away from zero is far enough to reject H ? That's why we need the distribution of T when H is true. [picture of T distribution] For example, when H 0 is true, and we have n=11 observations, then Pr{-3.196 < T 0 < +3.196} = 0.01 (from T-tables) Thus I set my criteria to be: "reject H if |T | > 3.196" Then I only have a 0.01 probability of a type I error Thus to summarize the test procedure: Take a sample of n observations X 1 , X 2 , ..., X n Compute the sample average Compute the sample standard deviation Compute T If |T | > T ( α , n-1) then reject H "P-values" of tests We can actually report results 2 ways: 1. State α ahead of time, and report if we reject H or not. 2. After analysis, state the value of α which is on the border of reject and do not reject....
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This note was uploaded on 08/06/2008 for the course IE 410 taught by Professor Storer during the Fall '04 term at Lehigh University .

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lect2 - A Quick Review of Hypothesis Testing In this...

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