Question

A two-way table is shown for two groups, 1 and \(2,\) and two possible outcomes, A and B. In each case, (a) What proportion of all cases had Outcome \(\mathrm{A}\) ? (b) What proportion of all cases are in Group \(1 ?\) (c) What proportion of cases in Group 1 had Outcome \(\mathrm{B} ?\) (d) What proportion of cases who had Outcome \(\mathrm{A}\) were in Group \(2 ?\) $$\begin{array}{|l|cc|c|}\hline & \text { Outcome A } & \text { Outcome B } & \text { Total } \\ \hline \text { Group 1 } & 40 & 10 & 50 \\ \text { Group 2 } & 30 & 20 & 50 \\\\\hline \text { Total } & 70 & 30 & 100 \\\ \hline\end{array}$$

Short answer

a) 70% b) 50% c) 20% d) Approximately 42.857%

Step-by-step solution

Proportion with Outcome A

There are a total of 70 cases with Outcome A. As the total number of cases is 100, the proportion of all cases that had Outcome A is \(\frac{70}{100} = 0.7\) or 70%.

Proportion in Group 1

There are a total of 50 cases in Group 1. Thus, the proportion of all cases that are in Group 1 is \(\frac{50}{100} = 0.5\) or 50%.

Proportion of cases in Group 1 with Outcome B

There are 10 cases in Group 1 with Outcome B. Thus, the proportion of cases in Group 1 that had Outcome B is \(\frac{10}{50} = 0.2\) or 20%.

Proportion of cases with Outcome A in Group 2

There are 30 cases with Outcome A in Group 2. Hence, the proportion of cases that had Outcome A and were in Group 2 is \(\frac{30}{70} = 0.42857\) approximately or 42.857%.