The scores on the questionnaire are approximately normally distributed with a mean of 49.5 and a standard deviation of 14. The customers with scores in the bottom 15% are described as “risk averse”. What is the questionnaire score that separates customers who are considered risk averse from those who are not?

To find the questionnaire score that separates customers who are considered "risk averse" (i.e., the score below which the bottom 15% of the scores fall), we need to determine the score corresponding to the 15th percentile of the normal distribution.

### Steps:
1. **Identify the parameters:**
   - Mean (\(\mu\)) = 49.5
   - Standard deviation (\(\sigma\)) = 14
   - Percentile (P) = 15%

2. **Find the z-score that corresponds to the 15th percentile:**
   - For a standard normal distribution, the z-score corresponding to the 15th percentile is approximately \(-1.04\).

3. **Convert the z-score to the actual score using the formula:**
   \[
   X = \mu + z \times \sigma
   \]
   Substituting the values:
   \[
   X = 49.5 + (-1.04) \times 14
   \]

Now, let's calculate the score.The questionnaire score that separates customers who are considered "risk averse" from those who are not is approximately **34.94**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the score for the top 10% of customers?
2. How would the risk-averse threshold change if the standard deviation were 10 instead of 14?
3. What percentage of customers have scores above 60?
4. How do you calculate the z-score for any given percentile?
5. How does changing the mean affect the score distribution?

**Tip:** Understanding the z-score helps in determining how far a specific score is from the mean in terms of standard deviations.

Discover how to calculate the questionnaire score threshold separating risk-averse customers from others. This problem involves normal distribution and percentile calculations with a detailed step-by-step solution. Suitable for advanced high school students.

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