Question

20 for a given school year, a reporter has been told that the average teachers salary was $59,500 with a standard deviation of $17,200. the reporter also knows that teachers will be receiving raises of 3.25% for the next school year. what would the reporter write for the new average teachers salary and standard deviation? (a) mean $1934; standard deviation $559 (b) mean $59,500; standard deviation $17,200 (c) mean $59,500; standard deviation $17,759 (d) mean $61,434; standard deviation $17,200 (e) mean $61,434; standard deviation $17,759

Explanation:

Step1: Calculate new mean

If each teacher's salary is increased by \(3.25\%\), we use the formula for the mean of a scaled data - set. If \(X\) is the original salary variable with mean \(\mu = 59500\), and the new salary \(Y=(1 + 0.0325)X\).
The formula for the mean of \(Y\) is \(\mu_Y=(1 + 0.0325)\mu_X\).
Substitute \(\mu_X = 59500\) into the formula: \(\mu_Y=1.0325\times59500=61434\).

Step2: Calculate new standard deviation

If \(X\) is the original salary variable with standard deviation \(\sigma_X = 17200\), and the new salary \(Y=(1 + 0.0325)X\).
The formula for the standard deviation of \(Y\) is \(\sigma_Y=(1 + 0.0325)\sigma_X\) (since for \(Y = aX\), \(\sigma_Y=\vert a\vert\sigma_X\) and \(a = 1.0325>0\)).
Substitute \(\sigma_X = 17200\) into the formula: \(\sigma_Y=1.0325\times17200 = 17759\).

Answer:

E. Mean \(\$61,434\); Standard deviation \(\$17,759\)