# Step-by-Step Solution Given: - Normal distribution with mean (\(\mu\)) = **40** - Standard deviation (\(\sigma\)) = **10** --- ## a. Find the value of \( x \) such that it is exceeded by no more than 25% of the distribution values. **Explanation:** - "Exceeded by no more than 25%" means 25% is **above** \( x \), so \( x \) is at the **75th percentile**. ### Steps: 1. **Find the z-score** for the 75th percentile: - From the standard normal table, \( P(Z < z) = 0.75 \) - \( z \approx 0.674 \) 2. **Convert to x using the formula:** \[ x = \mu + z\sigma \] \[ x = 40 + 0.674 \times 10 = 40 + 6.74 = 46.74 \] **Final Answer (a):** \[ \boxed{46.74} \] --- ## b. Find the value of \( x \) such that it is exceeded by 85% of the distribution values. **Explanation:** - "Exceeded by 85%" means 85% is **above** \( x \), so \( x \) is at the **15th percentile**. ### Steps: 1. **Find the z-score** for the 15th percentile: - \( P(Z < z) = 0.15 \) - \( z \approx -1.036 \) 2. **Convert to x:** \[ x = 40 + (-1.036) \times 10 = 40 - 10.36 = 29.64 \] **Final Answer (b):** \[ \boxed{29.64} \] --- ## c. Determine two values of \( x \) for which the smaller has 5% of its distribution values below it and the larger has 5% of its distribution values above it. **Explanation:** - Smaller: 5% **below** (\( P(Z < z) = 0.05 \)), which is the **5th percentile**. - Larger: 5% **above** (\( P(Z < z) = 0.95 \)), which is the **95th percentile**. ### Steps: 1. **Find the z-scores:** - 5th percentile: \( z \approx -1.645 \) - 95th percentile: \( z \approx 1.645 \) 2. **Convert to x:** - Smaller: \[ x_1 = 40 + (-1.645) \times 10 = 40 - 16.45 = 23.55 \] - Larger: \[ x_2 = 40 + 1.645 \times 10 = 40 + 16.45 = 56.45 \] **Final Answer (c):** \[ \boxed{23.55 \text{ and } 56.45} \] --- ## **Summary Table** | Part | Percentile | \(z\)-score | \(x\) value | |------|------------|-------------|-------------| | a | 75th | 0.674 | 46.74 | | b | 15th | -1.036 | 29.64 | | c | 5th/95th | -1.645/1.645| 23.55, 56.45| Let me know if you need explanation or calculations for any step!