1. **Stating the problem:** We have a survey of 1000 students at University ABC, with 750 boys and 250 girls. Among smokers, 22.9% are boys and 4.5% are girls. We need to construct a tree diagram illustrating these events. 2. **Understanding the problem:** The tree diagram will show the breakdown of students by gender first, then by smoking status within each gender. 3. **Step 1: Define events and probabilities.** - Total students: 1000 - Boys: 750 (probability $P(B) = \frac{750}{1000} = 0.75$) - Girls: 250 (probability $P(G) = \frac{250}{1000} = 0.25$) 4. **Step 2: Smoking percentages among boys and girls.** - Percentage of smokers who are boys: 22.9% - Percentage of smokers who are girls: 4.5% 5. **Step 3: Calculate total smokers percentage.** - Total smokers percentage = $22.9\% + 4.5\% = 27.4\%$ 6. **Step 4: Calculate probability of smoking given gender.** - Probability a student is a smoker: $P(S) = 0.274$ - Probability a smoker is a boy: $P(B|S) = 0.229 / 0.274 \approx 0.836$ (not needed for tree but useful) 7. **Step 5: Calculate smoking rates within boys and girls.** - Number of smokers who are boys: $229$ (22.9% of 1000) - Number of boys: 750 - Probability a boy smokes: $P(S|B) = \frac{229}{750} \approx 0.3053$ - Number of smokers who are girls: $45$ (4.5% of 1000) - Number of girls: 250 - Probability a girl smokes: $P(S|G) = \frac{45}{250} = 0.18$ 8. **Step 6: Construct the tree diagram structure:** - First branch: Gender with probabilities $P(B) = 0.75$, $P(G) = 0.25$ - Second branch: Smoking status given gender: - For boys: $P(S|B) = 0.3053$, $P(\text{non-smoker}|B) = 1 - 0.3053 = 0.6947$ - For girls: $P(S|G) = 0.18$, $P(\text{non-smoker}|G) = 0.82$ 9. **Summary:** The tree diagram starts with gender split, then each gender splits into smoker and non-smoker with the above probabilities. This completes the construction of the tree diagram illustrating the events.