Question

All 10,000 California students in the beginning of 8th grade are given an entrance exam that will allow them to attend a top academic charter school for free. Students who achieve a score of 92 or greater are admitted. This year the mean on the entrance exam was an 82 with a standard deviation of 4.5. a.What is the percentage of students who have the chance to attend the charter school

Answer 1

1.32% of students have the chance to attend the charter school.

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 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.

This year the mean on the entrance exam was an 82 with a standard deviation of 4.5.

This means that
`\mu = 82, \sigma = 4.5`

a.What is the percentage of students who have the chance to attend the charter school?

Students who achieve a score of 92 or greater are admitted, which means that the proportion is 1 subtracted by the pvalue of Z when X = 92. So

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

`Z = (92 - 82)/(4.5)`

`Z = 2.22`

`Z = 2.22` has a pvalue of 0.9868

1 - 0.9868 = 0.0132

0.0132*100% = 1.32%

1.32% of students have the chance to attend the charter school.