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 36 and the age of the winner for the best actress award was 54. for all recipients of best actor, the mean age is 44.6 years and the standard deviation is 7.5 years. for all recipients of best actress, the mean age is 36.8 years and the standard deviation is 12.4 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: Recall the z - score formula

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step2: Calculate the z - score for the Best Actor

For the Best Actor, \(x = 36\), \(\mu=44.6\), and \(\sigma = 8\).

$$ z_{Actor}=\frac{36 - 44.6}{8}=\frac{- 8.6}{8}=-1.075\approx - 1.08 $$

Step3: Calculate the z - score for the Best Actress

For the Best Actress, \(x = 54\), \(\mu = 36.8\), and \(\sigma=12.4\).

$$ z_{Actress}=\frac{54 - 36.8}{12.4}=\frac{17.2}{12.4}\approx1.39 $$

Step4: Compare the absolute values of the z - scores

The absolute value of \(z_{Actor}\) is \(|z_{Actor}| = 1.08\), and the absolute value of \(z_{Actress}\) is \(|z_{Actress}|=1.39\). Since \(1.39>1.08\), the Best Actress had the more extreme age.

Answer:

The z - score for the Best Actor is \(z=-1.08\), the z - score for the Best Actress is \(z = 1.39\), and the Best Actress had the more extreme age.