Question

The SAT is an exam that is used by many universities for admission. Suppose that the scores on the SAT mathematics exam have a normal distribution with mean 500 and standard deviation of 100. The statistics department identified students scoring in the top 4% of the SAT mathematics exam for recruitment. About what is the cutoff score for recruitment by the statistics department

Answer 1

The cutoff score for recruitment by the statistics department is 675.

Explanation:

Problems of normally distributed samples can be 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.

In this problem, we have that:

`\mu = 500, \sigma = 100`

Cutoff score for the top 4%.

100-4 = 96th percentile, which is X when Z has a pvalue of 0.96. So X when Z = 1.75.

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

`1.75 = (X - 500)/(100)`

`X - 500 = 1.75*100`

`X = 675`

The cutoff score for recruitment by the statistics department is 675.