Question

Find the predicted Math score percentile for a student with a 90th percentile Verbal score, given a correlation of 0.57 between SAT Verbal and Math scores.

Answer 1

To find the predicted Math score percentile for a student with a 90th percentile Verbal score, we can apply the concept of regression to the mean using the correlation between the two test scores.

Let's denote:

- \( P_V = 90 \) (the percentile for the Verbal score)
- \( r = 0.57 \) (the correlation between Verbal and Math scores)

### Steps to Calculate the Predicted Math Score Percentile

1. **Convert the Verbal Percentile to a Z-score**: The Z-score corresponding to the 90th percentile can be found using a Z-table or standard normal distribution table. The Z-score for the 90th percentile is approximately:

\[
Z_V = 1.28
\]

2. **Calculate the predicted Z-score for Math**: Using the linear regression formula considering the correlation:

\[
Z_M = r \cdot Z_V
\]

Substitute the values:

\[
Z_M = 0.57 \cdot 1.28 \approx 0.7296
\]

3. **Convert the Math Z-score back to a Percentile**: To find the percentile corresponding to the calculated Z-score of approximately 0.7296, we can reference the standard normal distribution table. The cumulative probability (percentile) for \( Z = 0.7296 \) is approximately 0.7673, or 76.73%.

### Conclusion

Thus, the predicted Math score percentile for a student with a 90th percentile Verbal score is approximately:

\[
\text{Predicted Math Score Percentile} \approx 76.73
\]

### Final Answer

**The predicted Math score percentile is approximately 77th percentile.**