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 34 and the age of the winner for the best actress award was 42. for all recipients of best actor, the mean age is 45.3 years and the standard deviation is 6.8 years. for all recipients of best actress, the mean age is 31.5 years and the standard deviation is 10.3 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 best actress, best actor had the more extreme age. (round to two decimal places.)

Explanation:

Step1: Calculate z - score for Best Actor

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For the Best Actor, $x = 34$, $\mu=45.3$, and $\sigma = 6.8$.
$z_{actor}=\frac{34 - 45.3}{6.8}=\frac{- 11.3}{6.8}\approx - 1.66$

Step2: Calculate z - score for Best Actress

For the Best Actress, $x = 42$, $\mu = 31.5$, and $\sigma=10.3$.
$z_{actress}=\frac{42-31.5}{10.3}=\frac{10.5}{10.3}\approx1.02$

Step3: Compare the absolute values of z - scores

The absolute value of $z_{actor}$ is $|z_{actor}|=1.66$, and the absolute value of $z_{actress}$ is $|z_{actress}| = 1.02$. Since $1.66>1.02$, the Best Actor has a more extreme age relative to their category.

Answer:

Since the z - score for the winner of Best Actor is $z=-1.66$ and the z - score for the winner of Best Actress is $z = 1.02$, the winner of Best Actor had the more extreme age.