Science scores for high school seniors in the United States are normally distributed with a mean of 78 and a standard deviation of 10. Students scoring in the top 10% are eligible for a special prize consisting of a laptop and $2,000. What is the approximate cutoff score a student must get in order to receive the prize?
To find the approximate cutoff score a student must get in order to receive the prize, we need to determine the z-score corresponding to the top 10% of the distribution, and then convert that z-score back to the original score using the mean and standard deviation.
1. Calculate the z-score corresponding to the top 10%:
Since the distribution is normal, we know that roughly 10% of scores fall above the mean.
Therefore, we need to find the z-score associated with the area to the right of the mean being 10%.
We can use a standard normal distribution table or a calculator to find the z-score.
The z-score corresponding to the top 10% is approximately 1.28.
2. Convert the z-score back to the original score:
To convert the z-score back to the original score, we can use the formula:
raw score = (z-score * standard deviation) + mean
Substituting the values we have:
raw score = (1.28 * 10) + 78
raw score = 12.8 + 78
raw score = 90.8
Therefore, the approximate cutoff score a student must get in order to receive the prize is 90.8.