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 \(2 ?\) $$\begin{array}{|l|cc|c|} \hline & \text { Outcome A } & \text { Outcome B } & \text { Total } \\\\\hline \text { Group 1 } & 20 & 80 & 100 \\ \text { Group 2 } & 60 & 40 & 100 \\\\\hline \text { Total } & 80 & 120 & 200 \\\ \hline\end{array}$$

Short answer

The proportions are as follows: (a) 0.4 (b) 0.5 (c) 0.8 (d) 0.75

Step-by-step solution

(a) Proportion of all cases with Outcome A

Look for the total number of cases with Outcome A which is 80. The total number of cases is 200. To find the proportion, divide the cases of Outcome A by the total cases, so it is \(\frac{80}{200} = 0.4\).

(b) Proportion of all cases in Group 1

Look for the total number of cases in Group 1 which is 100. As before, the total number of cases is 200. To find the proportion, divide the cases of Group 1 by the total cases, so it is \(\frac{100}{200} = 0.5\).

(c) Proportion of cases in Group 1 with Outcome B

Find the number of cases in Group 1 with Outcome B which is 80. The total number of cases in Group 1 is 100. To find the proportion, divide the cases in Group 1 with Outcome B by the total cases in Group 1, so it is \(\frac{80}{100} = 0.8\).

(d) Proportion of cases with Outcome A in Group 2

Find the number of cases in Group 2 with Outcome A which is 60. As before, the total number of cases with outcome A is 80. To find the proportion, divide the cases in Group 2 with Outcome A by the total cases with Outcome A, so it is \(\frac{60}{80} = 0.75\).