Question

1. In order to test the treatment A is better than B, we conduct an experiment and obtain the following table Success Failure Trt A 4 Trt B 2 6 (a) Compute p-value of the Fisher's exact test (b) Compute p-value of approximated test based on CLT (i.e., z-test (c) Which one do you think appropriate? Justify your answer (*) Do NOT use chisq.test() function in R.

Answer 1

a)

Fisher's exact test

The one-tailed P value equals 0.0849

  The association between rows (groups) and columns (outcomes)
   is considered to be not quite statistically significant.

b)

p-value = 0.0339

For sample 1, we have that the sample size is N1 = 12, the number of favorable cases is X1 = 8, so then the sample proportion is pi = = 0.6667 For sample 2, we have that the sample size is N2 = 8, the number of favorable cases is X2 = 2, so = 2 =0.25 then the sample proportion is p2 = 8+2 -0.5 12+8 Х+X, The value of the pooled proportion is computed as p = Also, the given significance level is a = 0.05 (1) Null and A/ternative Hypotheses The following null and alternative hypotheses need to be tested: Ho:p1 = P2 На: рi > Р2 This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a right-tailed test is ze = 1.64. The rejection region for this right-tailed test is R = {z : z> 1.64} (3) Test Statistics The z-statistic is computed as follows: р — р2 0.6667 0.25 1.826 VР(1 - р)(1/пt1/n2) V0.5- (1 - 0.5)(1/12 1/8) (4) Decision about the null hypothesis 1.826 > ze = 1.64, it is then concluded that the null hypothesis is Since it is observed that z rejected. Using the P-value approach: The p-value is p = 0.0339, and since p 0.0339 < 0.05, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion pi is greater than p2, at the 0.05 significance level.