Question

Suppose SAT Critical Reading scores are normally distributed with a mean of 501 and a standard deviation of 110. A university plans to admit students whose scores are in the top 30%. What is the minimum score required for admission

Answer 1

The minimum score required for admission is 558.75.

Explanation:

When the distribution is normal, we use 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.

Suppose SAT Critical Reading scores are normally distributed with a mean of 501 and a standard deviation of 110.

This means that
`\mu = 501, \sigma = 110`

A university plans to admit students whose scores are in the top 30%. What is the minimum score required for admission

This is the 100 - 30 = 70th percentile, which is X when Z has a pvalue of 0.7. So X when Z = 0.525. So

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

`0.525 = (X - 501)/(110)`

`X - 501 = 0.525*110`

`X = 558.75`

The minimum score required for admission is 558.75.