This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Chapter 11 Hypothesis Testing 2 Correct acceptance of H pr(I= H  T=H ) = (1 – α ) Type I Error pr(I= H 1  T=H ) = α [aka size ] Type II Error pr(I= H  T=H 1 ) = β Correct acceptance of H 1 pr(I= H 1  T=H 1 ) = (1 – β ) [aka power ] I : Inference in favor of: H 1 : Alternative Hypothesis H : Null Hypothesis H : Null Hypothesis T : The Truth of the matter: H 1 : Alternative Hypothesis 3 • Before diving into the “nuts and bolts” details, let’s consider the “big picture” of statistical testing – Example from p. 394 4 Components of a statistical test: – Background assumptions – Hypotheses – Test statistic – Determine the critical bound(s) – Interpret/Report the results. 5 1. For the results of a test for the mean of a population to have any meaning, the drawings from the sample must be: – Independent – Identically distributed 2. The underlying population must be approximately normal • This assumption is frequently violated • But when we use and a fairly large n (e.g., n ≥ 20), the CLT usually makes this true. Background Assumptions X 6 • We usually form only two hypotheses – The null hypothesis: H – The alternative hypothesis: H 1 • In this class, H is always a very specific hypothesis about the precise value of some parameter – E.g., we might use: H : μ = 120 Hypotheses 7 Test Statistic • For a test of the mean, we will use , often in its standardized form as a t statistic – comes from an unbiased and consistent point estimator x n s x s x h x h μ μ = x 8 ← → Accept H Accept H 1 Critical Bound: 9 • A critical bound acts as a threshold: – If our data results in a number less than the critical bound, we accept the hypothesis with the smaller mean (in this case, H ) – If our data results in a number greater than or equal to the critical bound, we accept the hypothesis with the larger mean (in this case, H 1 ) Determining a Critical Bound 10 Determining a Critical Bound • We will adopt the common practice of using α to determine our critical boundary α = pr(decide in favor of H 1  H is true) – How much of a risk are we willing to take of incorrectly inferring that H 1 is true when H actually is? – No matter what we do, we will have to take some chance (i.e., α > 0) 11 Determining a Critical Bound • A common choice of α is .05. – This choice means that we will set our critical bound so that only 5% of the distribution of H is beyond it: • This can be determined using quantiles: • @qnorm(.95) = 1.64 • @qtdist(.95, 19) = 1.73 • @qtdist(.95, @obs(x)1) = 1.73 95 . ) 64 . 1 ( = Φ df = n – 1 12 Only α of H ’s distribution (1 – α ) of H ’s distribution μ H0 ← → Accept H Accept H 1 Critical Bound: 1.64, if: 2200 α = .05 • n= 1, and • H = N(0, 1) 13 Determining a Critical Bound • Intuitively, we want to find the number b such that: • Similar to confidence intervals, we use α to determine a quantile q which then determines the boundary b 05 .  = ≥ H q S X pr X H μ ( 29 05 .  = ≥ H b X pr 14...
View
Full
Document
This note was uploaded on 10/01/2010 for the course ELEC 6111 taught by Professor Brown during the Spring '10 term at E. Illinois.
 Spring '10
 brown

Click to edit the document details