Question

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether patients getting Treatment \(\mathrm{A}\) are more likely to survive, if 63 out of 82 getting Treatment A survive and 31 out of 67 getting Treatment B survive.

Short answer

Using obtained sample and pooled proportions perform hypothesis test. If resulting p-value is less than significance level (typically 0.05), then, we reject the null hypothesis, i.e it is likely that patients getting Treatment A are more likely to survive than those getting Treatment B.

Step-by-step solution

Calculate the Relevant Sample Proportions

We have two treatments given, A and B. The proportion of patients surviving for Treatment A is calculated as the number of success cases divided by the sample size. Thus, \( p_A = 63 / 82 \). Likewise, the proportion of patients surviving for Treatment B can be calculated as \( p_B = 31 / 67 \).

Calculate the Pooled Proportion

The pooled proportion is found by adding the number of successful cases for both groups and dividing by the combined sample size. Therefore, \( p_{pooled} = (63 + 31) / (82 + 67) \).

Perform the Hypothesis Test

The null hypothesis \( H_0 \) is that success rates for treatments A and B are equal. The alternative hypothesis \( H_A \) is that treatment A has a higher success rate than treatment B. In other words, \( H_0: p_A = p_B \) and \( H_A: p_A > p_B \). The z-score can be calculated using the formula \( z = (p_A - p_B) / \sqrt{ p_{pooled} * (1 - p_{pooled}) * [(1 / n_A) + (1 / n_B)] } \) where \( n_A \) and \( n_B \) are the sample sizes for treatments A and B respectively. Once the z-score is calculated, the p-value can be found from standard normal tables.