Question

5. A study investigated survival rates for in-hospital patients who suffered cardiac arrest. Among 58, 593 patients who had cardiac arrest during the day, 11,604 survived and were discharged. Among 28,155 patients who suffered cardiac arrest at night, 4139 survived and were discharged. Use a o.o1 significance level to test the claim that survival rates are the same for day and night patients.

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 58593, the number of favorable cases is X1 = 11604 and the sample proportion is p^1​=X1/N1​=11604/58593​=0.198 (b) Sample 2 - The sample size is N2 = 28155, the number of favorable cases is X2 = 4139 and the sample proportion is p^2​=X2/N2​=4139/28155​=0.147 and the significance level is α=0.01 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 Two-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=0.01, therefore the critical value for this Two-tailed test is Zc​=2.5758. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>2.5758 i.e. Z>2.5758 or Z<-2.5758 (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|>18.2609)=0 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=18.2609 > Zc​=2.5758, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0, and since p=0≤0.01, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p1 is different than p2, at the 0.01 significance level.

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