To solve this problem, we’ll use the information provided to construct a Venn diagram based on three sets:

1. A: Students who think the teacher is funny.
2. B: Students who laugh at the teacher’s jokes to be polite.
3. C: Students who are excited about mathematics.

Given Information

• Total students = 200

• Individual Counts:

  • $|C| = 83$: Students excited about math.
  • $|A| = 80$: Students who think the teacher is funny.
  • $|B| = 112$: Students who laugh at the teacher’s jokes to be polite.

• Overlapping Counts:

  • $|B \cap C| = 48$: Students excited about math and laugh to be polite.
  • $|A \cap B| = 35$: Students think the teacher is funny and laugh to be polite.
  • $|A \cap C| = 22$: Students think the teacher is funny and are excited about math.
  • $|A \cap B \cap C| = 5$: Students who think the teacher is funny, laugh to be polite, and are excited about math.

Steps to Complete the Venn Diagram

1. Set the Title: For example, "Student Responses to Teacher Survey on Humor and Math."

2. Label the Sets:

   • Label Set $A$ as "Think teacher is funny."
   • Label Set $B$ as "Laugh to be polite."
   • Label Set $C$ as "Excited about math."

3. Fill in the Overlaps:

   • Start with the intersection of all three sets, $|A \cap B \cap C| = 5$.
   • For $A \cap B$ but not $C$: $|A \cap B| - |A \cap B \cap C| = 35 - 5 = 30$.
   • For $A \cap C$ but not $B$: $|A \cap C| - |A \cap B \cap C| = 22 - 5 = 17$.
   • For $B \cap C$ but not $A$: $|B \cap C| - |A \cap B \cap C| = 48 - 5 = 43$.

4. Fill in the Unique Sections for Each Set:

   • Students in only $A$: $|A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 80 - (30 + 17 + 5) = 28$.
   • Students in only $B$: $|B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 112 - (30 + 43 + 5) = 34$.
   • Students in only $C$: $|C| - (|A \cap C| + |B \cap C| - |A \cap B \cap C|) = 83 - (17 + 43 + 5) = 18$.

5. Legend and Labels:

   • Include a legend explaining each category, with labels for each set $A$, $B$, and $C$, and the numbers assigned to each region.

Would you like a detailed visual breakdown of how the numbers fit into each section of the Venn diagram, or any other specific details?

Related Questions

1. How would the Venn diagram change if another category, such as "not interested in math," were added?
2. Can you verify the total number of students by adding all individual and overlapping groups?
3. What probability is there for a student chosen at random to fall into each of the regions?
4. How would the Venn diagram look if the survey included more students?
5. Could we calculate conditional probabilities, such as the probability a student thinks the teacher is funny given that they laugh to be polite?

Tip

When working with Venn diagrams, it’s helpful to start with the most specific intersections and work outward to avoid double-counting.