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Lab 8 STA 570 401-404 Sample Size calculations Objectives : To practice computing sample sizes required to achieve particular experimental design criteria, such as type I and power values. To start : Log in to the computers and start Word. We will not be using SAS for this lab. Remember the formula from lecture for sample size determinations 2 z1s1 z0 s0 n p0 p1 With your TA 1) Suppose we wish to test H0 : p=0.4 against H1 : p>0.4 using α=0.01. We also wish to achieve 80% power when p=0.42. What is the minimal required sample size? Here we have a “greater than” alternative, so we need to equate the 1-α=0.99 percentile of the null distribution to the 1-POW=0.20 percentile of the alternative distribution. Thus, z0=2.33, z1=(-0.84), s0=sqrt(0.4*0.6)=0.4899, and s1=sqrt(0.42*0.58)=0.4936. The required sample size is ( 084)(0.4936) (2.33)(0.4899) 2 . n 6053548 . 0.4 0.42 2) Suppose we wish to test H0 : p=0.3 against H1 : p≠0.3. We wish to use α=0.05 and achieve 90% power with p=0.35. What is the minimal required sample size? Here will have a “not equal” alternative with a p1 greater than p0. Thus we need to equate the 1-(α/2)=0.975 percentile of the null distribution to the 1-POW=0.10 percentile of the alternative distribution. Thus z0=1.96, z1=(-1.28), s0=sqrt(0.3*0.7)=0.4583, and s1=sqrt(0.35*0.65)=0.4770. Thus the minimal required sample size is ( 128)(0.4770) (196)(0.4583) 2 . . n 910.625 0.3 0.35 On your own 1) You are testing a roulette wheel. In theory, the roulette wheel should produce 18/38 (0.4737) black spins. You want to test whether or not the wheel actually produces this probability (erring on either side is bad). You want to use α=0.10 and want to achieve 99% power when p=0.45. What is the minimal required sample size? 2) You want to test whether a counseling program for married couples reduces the divorce rate below 30%. You will enroll n couples in the program and observe how many still get divorced. You want to have the type I error probability equal to 0.05 and achieve 90% power if the program reduces the divorce rate to 25%. What is the minimal number of couples required?