\( 1 \leftarrow \quad \begin{array}{l}\text { A standardized exam's scores are normally distributed In a recont year, the mean test score was } 1525 \text { and the } \\ \text { standard deviation was 316. The test scores of four students selected at randorn are } 1950,1290,2250 \text {, and } 1410 \text {. } \\ \text { Find the } z \text { scores that correspond to each value and determine whether any of the values are unusual }\end{array} \)

To determine whether any of the student test scores are unusual, we'll first calculate the **z-scores** for each score. A **z-score** indicates how many standard deviations a data point is from the mean. Typically, a score is considered **unusual** if its z-score is **greater than 2** or **less than -2**. ### Given: - **Mean (\( \mu \))** = 1525 - **Standard Deviation (\( \sigma \))** = 316 - **Test Scores**: 1950, 1290, 2250, 1410 ### Z-Score Formula: \[ z = \frac{{X - \mu}}{{\sigma}} \] ### Calculations: 1. **Score: 1950** \[ z = \frac{{1950 - 1525}}{{316}} = \frac{{425}}{{316}} \approx 1.34 \] 2. **Score: 1290** \[ z = \frac{{1290 - 1525}}{{316}} = \frac{{-235}}{{316}} \approx -0.74 \] 3. **Score: 2250** \[ z = \frac{{2250 - 1525}}{{316}} = \frac{{725}}{{316}} \approx 2.30 \] 4. **Score: 1410** \[ z = \frac{{1410 - 1525}}{{316}} = \frac{{-115}}{{316}} \approx -0.36 \] ### Summary of Z-Scores: - **1950**: \( z \approx 1.34 \) - **1290**: \( z \approx -0.74 \) - **2250**: \( z \approx 2.30 \) - **1410**: \( z \approx -0.36 \) ### Determining Unusual Scores: - **Unusual Criterion**: \( |z| > 2 \) - **1950**: \( |1.34| < 2 \) → **Not Unusual** - **1290**: \( |-0.74| < 2 \) → **Not Unusual** - **2250**: \( |2.30| > 2 \) → **Unusual** - **1410**: \( |-0.36| < 2 \) → **Not Unusual** ### **Conclusion:** Only the test score of **2250** is **unusual** as its z-score of **2.30** exceeds the typical threshold of **2** standard deviations from the mean.