Question

use z scores to compare the given values.
in a recent awards ceremony, the age of the winner for the best actor award was 35 and the age of the winner for the best actress award was 55. for all recipients of best actor, the mean age is 45.4 years and the standard deviation is 8.7 years. for all recipients of best actress, the mean age is 33.5 years and the standard deviation is 12.3 years. (all ages are determined at the time of
the awards ceremony.) relative to the award category, who had the more extreme age when winning the award, the winner of best actor or the winner of best actress? explain.
since the z score for the winner of best actor is z = □ and the z score for the winner of best actress is z = □, the winner of □ had the more extreme age.
(round to two decimal places.)

Explanation:

Step1: Calculate the z - score for Best Actor

The formula for the z - score is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For Best Actor: \(x = 35\), \(\mu=45.4\), \(\sigma = 8.7\)
\(z=\frac{35 - 45.4}{8.7}=\frac{- 10.4}{8.7}\approx - 1.20\)

Step2: Calculate the z - score for Best Actress

For Best Actress: \(x = 55\), \(\mu = 33.5\), \(\sigma=12.3\)
\(z=\frac{55 - 33.5}{12.3}=\frac{21.5}{12.3}\approx1.75\)

Step3: Compare the absolute values of z - scores

The absolute value of the z - score for Best Actor is \(|z_{Actor}|=1.20\)
The absolute value of the z - score for Best Actress is \(|z_{Actress}| = 1.75\)
Since \(1.75>1.20\)

Answer:

The z - score for the winner of Best Actor is \(z=-1.20\) and the z - score for the winner of Best Actress is \(z = 1.75\). The winner of Best Actress had the more extreme age.