The scores on a test are normally distributed with a mean of 40 and a standard deviation of 8. What is the score that is \( 2 \frac{1}{2} \) standard deviations above the mean? A score of \( \square \) is \( 2 \frac{1}{2} \) standard deviations above the mean.

To find a score that is \( 2 \frac{1}{2} \) standard deviations above the mean, first convert \( 2 \frac{1}{2} \) to an improper fraction, which is \( 2.5 \). Then, calculate the score using the formula: \[ \text{Score} = \text{Mean} + (Z \times \text{Standard Deviation}) \] Substituting in the values gives us: \[ \text{Score} = 40 + (2.5 \times 8) = 40 + 20 = 60 \] So, a score of \( 60 \) is \( 2 \frac{1}{2} \) standard deviations above the mean. Apart from its mathematical properties, this concept is widely applicable in fields like psychology or education where understanding performance distribution helps educators tailor their teaching. Knowing the score necessary to exceed a specific percentile can help set realistic goals for student achievement! Additionally, a common mistake is forgetting to account for whether you are dealing with standard deviations above or below the mean when calculating your new score. Always clarify if you’re looking for above or below the mean before jumping into calculations—precision is key in statistics!