Question

in a random sample of 54 patients undergoing a standard surgical procedure,10 required pain medication. in a random sample of 80 patients undergoing a new procedure, 19 required pain medication. test the claim that few pain medication for the new procedure as compared to the standarfd procedure at the 5% significance level.

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 54, the number of favorable cases is X1 = 10 and the sample proportion is p^1​=X1/N1​=10/54​=0.1852 (b) Sample 2 - The sample size is N2 = 80, the number of favorable cases is X2 = 19 and the sample proportion is p^2​=X2/N2​=19/80​=0.2375 and the significance level is α=0.05 Pooled Proportion The value of the pooled proportion is computed as (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = p2 Ha: p1 > p2 This corresponds to a Right-tailed test, for which a z-test for two population proportions needs to be conducted. (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Right-tailed test is Zc​=1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Right-tailed test is Z>1.6449 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(Z>-0.7213)=0.7646 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that Z=-0.7213 < Zc​=1.6449, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.7646, and since p=0.7646>0.05, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population proportion p1 is greater than p2, at the 0.05 significance level.

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