Hypothesis Testing in Statistics: Errors, Decision Rules, Power, and Sample Size, Exams of Data Analysis & Statistical Methods

Download Hypothesis Testing in Statistics: Errors, Decision Rules, Power, and Sample Size and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! STATISTICS 571 TA: Perla Reyes DISCUSSION 7 Review 1. Errors in Hypothesis Testing H0 is true H0 is false Reject H0 Type I error OK Accept H0 OK Type II error (a) α=P (Type I error)=P (reject H0 when H0 is true) (b) β=P (Type II error)=P (accept H0 for a particular alternative value) (c) The power is the probability of rejecting H0 given that the true value of the parameter being tested is some specified value. power= 1− β 2. Decision Rule and Rejection Region (a) One-sided test H0 : µ = µ0, vs HA : µ > µ0 For an α level, i. The rejection region is X̄ ≥ Zα σ√n + µ0 ii. Let be c = Zα σ√n + µ0 the point that marks the rejection region. iii. The decision rule is: Reject H0 if x̄ ≥ c (b) Two-sided test H0 : µ = µ0, vs HA : µ 6= µ0 For an α level, i. The rejection region is X̄ ≤ −Zα/2 σ√n + µ0 OR X̄ ≥ Zα/2 σ√ n + µ0 ii. Let be c1 = −Zα/2 σ√n + µ0 and c2 = Zα σ/2√ n + µ0 the points that mark the rejection region. iii. The decision rule is: Reject H0 if x̄ ≤ c1 OR x̄ ≥ c2 Note: P (Z ≥ Zα) = α, example Z0.025 = 1.96 3. Power Determination (a) One-sided test H0 : µ = µ0, vs HA : µ > µ0 If my rejection region is X̄ ≥ c, and I want to know the power of my test when the real value µ = µA. power = P (reject H0|µ = µA) = P (X̄ ≥ c|µ = µA) power = P (Z ≥ c−µA σ/ √ n ) (b) Two-sided test H0 : µ = µ0, vs HA : µ 6= µ0 If my rejection region is X̄ ≤ c1 OR X̄ ≥ c2 , and I want to know the power of my test when the real value µ = µA. power = P (reject H0|µ = µA) = P (X̄ ≤ c1|µ = µA) + P (X̄ ≥ c2|µ = µA) power = P (Z ≤ c1−µA σ/ √ n ) + P (Z ≥ c2−µA σ/ √ n ) 4. Sample size determination (a) One-sided test H0 : µ = µ0, vs HA : µ > µ0 Suppose we want level equal to α, and power equal 1−β when the real µ = µA. In this case I need to know the variance of the population σ2. Let c be the point that marks the rejection region. email: reyes@stat.wisc.edu 1 Office: 248 MSC M2:30-3:30 R3:30-4:30 STATISTICS 571 TA: Perla Reyes DISCUSSION 7 P (X̄ ≥ c|µ = µ0) = α, and P (X̄ ≥ c|µ = µA) = power. α = P (Z ≥ c−µ0 σ/ √ n ), then c−µ0 σ/ √ n = Zα, and c = Zα(σ/ √ n) + µ0. power = P (Z ≥ c−µA σ/ √ n ), then c−µA σ/ √ n = Zβ, and c = Zβ(σ/ √ n) + µA Making right sides of both equations to be equal to each other and solving for n, n = σ2 × (Zα + Zβ)2 (µHA − µHo)2 (b) Two-sided test H0 : µ = µ0, vs HA : µ 6= µ0 Suppose we want level equal to α, and power equal 1−β when the real µ = µA. In this case I need to know the variance of the population σ2. Let c1 and c2 be the points that mark the rejection region. P (X̄ ≤ c1|µ = µ0) = α/2, P (X̄ ≥ c2|µ = µ0) = α/2, and P (X̄ ≤ c1|µ = µA) + P (X̄ ≥ c2|µ = µA) = power. The algebra to determine n appears to be more complicated, but since the contribution of one of the tails to the power is very small, we can approximate the power value with just one tail. We can focus on conditions just for c2. α/2 = P (Z ≥ c2−µ0 σ/ √ n ), then c2−µ0 σ/ √ n = Zα/2, and c2 = Zα/2(σ/ √ n) + µ0. power = P (Z ≥ c2−µA σ/ √ n ), then c2−µA σ/ √ n = Zβ, and c2 = Zβ(σ/ √ n) + µA Making right sides of both equations to be equal to each other and solving for n, n = σ2 × (Zα 2 + Zβ)2 (µHA − µHo)2 (c) (1− α)100% confidence interval As we saw in the previos discussion, let W be the width of the interval, n = ( 2× Zα/2 × σ W )2 Practice Problem 1. A machine fills milk bottles, the mean amount of milk in each bottle is supposed to be 32 Oz with a standard deviation of 0.06 Oz. Suppose the mean amount of milk is approximately normally distributed. To check if the machine is operating properly, 36 filled bottles will be chosen at random and the mean amount will be determined. (a) If an α = .05 test is used to decide whether the machine is working properly, what should the rejection criterion be? (b) Find the power of the test if the true mean takes on the following values: 31.97, 31.99, 32, 32.01, 32.03. Draw the power curve. email: reyes@stat.wisc.edu 2 Office: 248 MSC M2:30-3:30 R3:30-4:30