Suppose that the annual rate of return for a common biotechnology stock is normally distributed with a mean of 4% and a standard deviation of 6. Find the probability that the one-year retum of this stock will be negative. Round your answer to at least four decimal places.

To find the probability that the one-year return of the stock will be negative, we need to determine the probability that the return is less than 0%. Since the return is normally distributed, we can use the Z-score formula.

### Steps:

1. **Convert the raw score (0%) to a Z-score**:
   The Z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X \) is the raw score (0% in this case),
   - \( \mu \) is the mean (4%),
   - \( \sigma \) is the standard deviation (6%).

   Substituting the values:
   \[
   Z = \frac{0\% - 4\%}{6\%} = \frac{-4\%}{6\%} = -\frac{2}{3} \approx -0.6667
   \]

2. **Find the cumulative probability** for the Z-score:
   We need to find \( P(Z \leq -0.6667) \), which represents the probability that the return is less than 0%.

   Using a Z-table or a calculator, the cumulative probability for \( Z = -0.6667 \) is approximately 0.2525.

Thus, the probability that the one-year return of this stock will be negative is approximately **0.2525**.

Would you like more details or have any questions about this solution?

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Here are 5 related questions:

1. What is the probability that the one-year return of the stock will be more than 10%?
2. How would the probability change if the mean return were 6% instead of 4%?
3. What is the probability that the return will be between 2% and 8%?
4. How would you calculate the probability if the return distribution were not normal?
5. What is the Z-score for a return of -2%?

**Tip:** When dealing with normally distributed financial data, understanding how to calculate and interpret Z-scores helps assess the risk and likelihood of various returns.

Learn how to calculate the probability of a negative one-year return for a biotechnology stock using normal distribution and Z-scores. Get a step-by-step solution with a mean of 4% and a standard deviation of 6%. Approximate the probability to four decimal places.

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