Question

P1 To test the hypothesis Ho: p = 1/2 against H :p < 1/2, we take a random sample of Bernoulli trials, X,X,..., X.,, and use for our test statistic Y = ¿x, which has a binomial distribution b(n, p). Let the critical region be defined by C ={y:y sc}. Find the values of n and c so that (approximately) a = 0.05 and B = 0.10 when p = 1/4. 1=1

Answer 1

1 1 Here we want to test the null hypothesis Ho: p= - against the alternative H :p 2 2 We have the critical region y Sc, where y follows binomial distribution with parameter n and p 1 We have a = 0.05 i.e. probability rejecting null hypothesis when p 12. Plyselp-1)-(1-1) = 0.05 -(1) Also we have, 1 1-B = 0.90 when p = i.e. probability of rejecting null hypothesis when p= 4 ic. P[ysel p-1)-CH(-) = 0.90 By solving (1) and (ii) together we get n and c. For this we can use cumulative binomial probability table. p=(1/2) p=(1/4) n 10 15 20 25 30 10 15 20 25 30 1 0.0107 0.0005 0.0000 0.0000 0.0000 0.2440 0.0802 0.0243 0.0070 0.0020 2 0.0547 0.0037 0.0002 0.0000 0.0000 0.5256 0.2361 0.0913 0.0321 0.0106 3 0.1719 0.0176 0.0013 0.0001 0.0000 0.7759 0.4613 0.2252 0.0962 0.0374 4 0.3770 0.0592 0.0059 0.0005 0.0000 0.9219 0.6865 0.4148 0.2137 0.0979 5 0.6230 0.1509 0.0207 0.0020 0.0002 0.9803 0.8516 0.6172 0.3783 0.2026 6 0.8281 0.3036 0.0577 0.0073 0.0007 0.9965 0.9434 0.7858 0.5611 0.3481 7 0.9453 0.5000 0.1316 0.0216 0.0026 0.9996 0.9827 0.8982 0.7265 0.5143 8 0.9893 0.6964 0.2517 0.0539 0.0081 1.0000 0.9958 0.9591 0.8506 0.6736 9 0.9990 0.8491 0.4119 0.1148 0.0214 1.0000 0.9992 0.9861 0.9287 0.8034 10 1 0.9408 0.5881 0.2122 0.0494 1 0.9999 0.9961 0.9703 0.8943 11 0.9824 0.7483 0.3450 0.1002 1.0000 0.9991 0.9893 0.9493 12 0.9963 0.8684 0.5000 0.1808 1.0000 0.9998 0.9966 0.9784 13 0.9995 0.9423 0.6550 0.2923 1.0000 0.9991 0.9918 14 1.0000 0.9793 0.7878 0.4278 1.0000 0.9998 0.9973 15 1 0.9941 0.8852 0.5722 1 1 1.0000 0.9992 16 0.9987 0.9461 0.7077 1 1.0000 0.9998 17 0.9998 0.9784 0.8192 1 1.0000 0.9999 18 1.0000 0.9927 0.8998 1 1 1.0000 19 1.0000 0.9980 0.9506 1 1 1.0000 20 1 9995 09786 1 1 1 1 Form the above table, we get the approximate solution for n and c. And they are n= 30 and c = 10 approximately NOTE: to generate cumulative probability we can use the function=BINOM.DIST() in MS Excel