Question

The SAT and ACT college entrance exams are taken by thousands of students each year. The mathematics portions of each of these exams produce scores that are approximately normally distributed. In recent years, SAT mathematics exam scores have averaged 480 with standard deviation 100. The average and standard deviation for ACT mathematics scores are 18 and 6, respectively. (a) An engineering school sets 555 as the minimum SAT math score for new students. What percentage of students will score below 555 in a typical year? (Round your answer to two decimal places.)

Answer 1

77.34% of students will score below 555 in a typical year

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

SAT:

`\mu = 480, \sigma = 100`

(a) An engineering school sets 555 as the minimum SAT math score for new students. What percentage of students will score below 555 in a typical year?

This is the pvalue of Z when X = 555. So

`Z = (X - \mu)/(\sigma)`

`Z = (555 - 480)/(100)`

`Z = 0.75`

`Z = 0.75` has a pvalue of 0.7734.

77.34% of students will score below 555 in a typical year