Let's break down and solve the problem step by step. --- ## **Step 1: Understanding the** You are: - **Mean test score (\(\mu\))**: 202 - **Standard deviation (\(\sigma\))**: 38.2 You are asked to **find the test score corresponding to the following z-scores** (rounded to the nearest whole number): - \( z = 2.50 \) - \( z = -2.50 \) --- ## **Step 2: Z-Score Formula** The formula to convert a z-score to a raw score (test score) is: \[ X = \mu + (z \times \sigma) \] Where: - \( X \) = test score - \( \mu \) = mean - \( z \) = z-score - \( \sigma \) = standard deviation --- ## **Step 3: Calculations** ### **For \( z = 2.50 \):** \[ X = 202 + (2.50 \times 38.2) \] \[ X = 202 + 95.5 \] \[ X = 297.5 \] Rounded to the nearest whole number: \[ \boxed{298} \] --- ### **For \( z = -2.50 \):** \[ X = 202 + (-2.50 \times 38.2) \] \[ X = 202 - 95.5 \] \[ X = 106.5 \] Rounded to the nearest whole number: \[ \boxed{107} \] --- ## **Step 4: Final Answers** - **For \( z = 2.50 \): Test score = 298** - **For \( z = -2.50 \): Test score = 107** --- ### **Summary Table** | Z-score | Test Score (rounded) | |---------|----------------------| | 2.50 | 298 | | -2.50 | 107 | --- **These are the test scores corresponding to the given z-scores, rounded to the nearest whole number.**