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The problem you're working on involves finding the score that separates the bottom 5% of customers considered "risk averse" from those who are not. The scores are normally distributed with a mean of 51 and a standard deviation of 15.

### Steps to Solve:
1. **Find the z-score corresponding to the bottom 5%**:
   - Use a z-table or a calculator to find the z-score for 5% in the left tail of a normal distribution. The z-score corresponding to 5% is approximately -1.645.

2. **Apply the z-score formula** to convert the z-score to the corresponding questionnaire score:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   Where:
   - \( z = -1.645 \) (from step 1),
   - \( \mu = 51 \) (mean),
   - \( \sigma = 15 \) (standard deviation),
   - \( x \) is the score we are solving for.

3. **Solve for \( x \)**:
   \[
   x = z \cdot \sigma + \mu
   \]
   Substituting the values:
   \[
   x = (-1.645) \cdot 15 + 51
   \]
   \[
   x = -24.675 + 51 = 26.325
   \]
   Rounding to one decimal place, the questionnaire score is **26.3**.

Thus, the score that separates customers considered risk-averse from those who are not is **26.3**.

Let me know if you need further details!

Here are some related questions:
1. What is the significance of the z-score in this context?
2. How would the answer change if the standard deviation were smaller or larger?
3. What is the area under the normal curve to the left of the z-score of -1.645?
4. How would you find the cutoff for the top 5% of scores?
5. How does a change in the mean affect the cutoff score?

**Tip**: When using z-scores for real-world problems, it's essential to ensure you're referencing the correct tail of the distribution!

Risk taking is an important part of investing. In order to make suitable investment decisions on behalf of their customers, portfolio managers give a questionnaire to new customers to measure their desire to take financial risks. The scores on the questionnaire are approximately normally distributed with a mean of 51 and a standard deviation of 15. The customers with scores in the bottom 5% are described as 'risk averse.' What is the questionnaire score that separates customers who are considered risk averse from those who are not? Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.

Learn how to determine the questionnaire score that separates risk-averse customers based on a normal distribution with a mean of 51 and a standard deviation of 15. The calculation involves using the z-score formula and finding the cutoff for the bottom 5%.

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