Question

In 2017, 1,764,865 students took the SAT exam. The distribution of scores in the verbal section of the SAT had a mean µ = 467 and a standard deviation σ = 111. Let X = a SAT exam verbal section score in 2017. Then X ~ N(467, 111). Find the z-scores for x1 = 235 and x2 = 368.

Answer 1

X1 zscore

`Z = -2.09`

X2 zscore

`Z = -0.89`

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.

In this problem, we have that:

`\mu = 467, \sigma = 111`

x1 = 235

Z when
`X = 235`

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

`Z = (235 - 467)/(111)`

`Z = -2.09`

x2 = 368.

Z when
`X = 368`

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

`Z = (368 - 467)/(111)`

`Z = -0.89`