Question

The combined SAT scores for the students at a local high school are normally distributed with a mean of 1527 and a standard deviation of 295. The local college includes a minimum score of 1380 in its admission requirements. What percentage of students from this school earn scores that satisfy the admission requirement

Answer 1

69.15% of students from this school earn scores that satisfy the admission requirement.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 1527 and a standard deviation of 295.

This means that
`\mu = 1527, \sigma = 295`

The local college includes a minimum score of 1380 in its admission requirements. What percentage of students from this school earn scores that satisfy the admission requirement?

The proportion is 1 subtracted by the pvalue of Z when X = 1380. So

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

`Z = (1380 - 1527)/(295)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3085

1 - 0.3085 = 0.6915

0.6915*100% = 69.15%

69.15% of students from this school earn scores that satisfy the admission requirement.