Exam Pattern: Allocation Ratio Graph for Comparing Proportions
This note abstracts the makeup midterm question where a graph gives the total sample size needed for a target power, and the task is to recover the arm-specific sample size under a fixed allocation ratio.
Source Question
STA305 Makeup W2026, Question 4
Setup
Suppose the graph gives the total required sample size $N$ for a binary-outcome trial, and the desired allocation ratio is
$$ n_S : n_E = 3 : 1, $$where $n_S$ is the standard arm and $n_E$ is the experimental arm.
Workflow
Step 1: Read the total sample size from the graph
In the source problem, the graph gave approximately
$$ N \approx 575. $$Step 2: Translate the ratio into equations
Write
$$ n_S = 3n_E, \quad n_S + n_E = N. $$Step 3: Solve for the requested arm size
Substitute to get
$$ 3n_E + n_E = 575 \quad \Rightarrow \quad 4n_E = 575 \quad \Rightarrow \quad n_E \approx 143.75. $$So the experimental arm needs about
$$ 144 $$patients.
Exam Checklist
- Read the total sample size from the graph, not an arm size
- Convert the allocation ratio into algebra
- Solve for the requested arm, not just the total
- Round to a practical integer at the end