(a) To find Scott's percentile among the national group of test takers, we'll use the z-score formula:
\[z = \frac{(X - μ)}{σ}\]
Where:
- X is Scott's score (64).
- μ (mu) is the mean score for the national group (46.9).
- σ (sigma) is the standard deviation for the national group (10.9).
Calculating the z-score:
\[z = \frac{(64 - 46.9)}{10.9} \approx 1.39\]
Now, we can find Scott's percentile using the z-score. You can use a standard normal distribution table or calculator to find the percentile associated with a z-score of 1.39. The percentile will tell you how Scott's score compares to other test takers in the national group.
(b) To calculate Scott's z-score at his school, we'll use the same formula with the mean and standard deviation for his school:
\[z_{school} = \frac{(64 - 58.2)}{9.4} \approx 0.617\]
This z-score corresponds to Scott's performance relative to his school group.
Now, let's compare his z-scores:
- Scott's z-score among the national group is 1.39.
- Scott's z-score among his school group is 0.617.
(c) To assess how well the boys at Scott's school performed on the PSAT, we can use Scott's percentile among his school group. Scott's z-score of 0.617 corresponds to the 68th percentile, which means he scored higher than approximately 68% of the boys at his school. This suggests that the boys at Scott's school, as a group, performed reasonably well on the PSAT Math Section compared to a nationally representative group, as Scott's z-score is positive and his percentile is above 50%.
Hope this helps! :)