Question

A group of high school athletes has an average GPA of 2.7 with a standard deviation of 0.9. a)Find the percentage of athletes whose GPA more than 1.665. b) John's GPA is more than 85.31 percent of the athletes in the study. Compute his GPA.

Answer 1

a) The percentage of athletes whose GPA more than 1.665 is 87.49%.

b) John's GPA is 3.645.

Explanation:

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 = 2.7, \sigma = 0.9`

a)Find the percentage of athletes whose GPA more than 1.665.

This is 1 subtracted by the pvalue of Z when X = 1.665. So

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

`Z = (1.665 - 2.7)/(0.9)`

`Z = -1.15`

`Z = -1.15` has a pvalue of 0.1251

1 - 0.1251 = 0.8749

The percentage of athletes whose GPA more than 1.665 is 87.49%.

b) John's GPA is more than 85.31 percent of the athletes in the study. Compute his GPA.

His GPA is X when Z has a pvalue of 0.8531. So it is X when Z = 1.05.

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

`1.05 = (X - 2.7)/(0.9)`

`X - 2.7 = 1.05*0.9`

`X = 3.645`

John's GPA is 3.645.