Question

out of 20 students taking a midterm psychology exam, 13 answered the first of two bonus questions, 11 answered the second bonus question, and 1 didnt bother with either one. round your answer to the nearest percent.
(a) construct a venn diagram illustrating these results.

Explanation:

Step1: Find students who answered both

Let $A$ = first bonus responders, $B$ = second bonus responders.
Use principle: $|A \cap B| = |A| + |B| - |A \cup B|$.
First, $|A \cup B| = 20 - 1 = 19$.

$$\begin{align} |A \cap B| &= 13 + 11 - 19 \\ &= 5 \end{align}$$

Step2: Find only first bonus responders

Subtract both from first group:

$$\begin{align} |A| - |A \cap B| &= 13 - 5 \\ &= 8 \end{align}$$

Step3: Find only second bonus responders

Subtract both from second group:

$$\begin{align} |B| - |A \cap B| &= 11 - 5 \\ &= 6 \end{align}$$

Step4: Identify outside region

Given as 1 student who answered neither.

Answer:

• Universal set ($U$) outer box: total 20 students
• Only First bonus question circle: 8
• Overlap (both bonus questions): 5
• Only Second bonus question circle: 6
• Outside both circles (neither question): 1

To convert to percentages (rounded):

• Only First bonus: $\frac{8}{20} \times 100 = 40\%$
• Both bonus questions: $\frac{5}{20} \times 100 = 25\%$
• Only Second bonus: $\frac{6}{20} \times 100 = 30\%$
• Neither bonus question: $\frac{1}{20} \times 100 = 5\%$