Question

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $48,000 and a standard deviation of $5,400. We randomly survey ten teachers from that district. (Round your answers to the nearest dollar.) (a) Find the 90th percentile for an individual teacher's salary.

Answer 1

The 90th percentile for an individual teacher's salary is $54,912.

Explanation:

Answer 2

The 90th percentile for an individual teacher's salary is $54,912.

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 = 48000, \sigma = 5400`

Find the 90th percentile for an individual teacher's salary.

This is the value of X when Z has a pvalue of 0.9. So it is X when Z = 1.28.

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

`1.28 = (X - 48000)/(5400)`

`X - 48000 = 1.28*5400`

`X = 54912`

The 90th percentile for an individual teacher's salary is $54,912.