Question

The national collegiate athletic association (ncaa) uses a sliding scale for eligibility for division i athletes. those students with a 2.5 high school gpa must score at least 820 on the combined mathematics and critical reading parts of the sat to compete in their first college year. the combined scores of the almost 1.7 million high school seniors taking the sat in 2013 were approximately normal with mean 1011 and standard deviation 216. what percent of seniors score at least 820

Answer 1

81.06% of seniors score at least 820

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 = 1011, \sigma = 216`

What percent of seniors score at least 820

This is 1 subtracted by the pvalue of Z when X = 820. So

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

`Z = (820 - 1011)/(216)`

`Z = -0.88`

`Z = -0.88` has a pvalue of 0.1894

1 - 0.1894 = 0.8106

81.06% of seniors score at least 820

Answer 2

A suitable probability calculator pegs that probability at 81%.