Let's solve the given problem step-by-step using z-scores.
Given the information:
- The age of the Best Actor winner is 38 years.
- The mean age for Best Actor recipients is 46.4 years.
- The standard deviation for Best Actor recipients is 5.1 years.
1. Calculate the z-score for the Best Actor winner:
The formula for the z-score is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Where:
- [tex]\( X \)[/tex] is the value (age of the winner),
- [tex]\( \mu \)[/tex] is the mean age,
- [tex]\( \sigma \)[/tex] is the standard deviation.
For the Best Actor:
[tex]\[ X = 38 \][/tex]
[tex]\[ \mu = 46.4 \][/tex]
[tex]\[ \sigma = 5.1 \][/tex]
Substituting these values into the formula:
[tex]\[ z_{\text{actor}} = \frac{38 - 46.4}{5.1} \approx -1.65 \][/tex]
So, the z-score for the Best Actor winner is [tex]\(-1.65\)[/tex].
2. Next, let's calculate the z-score for the Best Actress winner:
Given:
- The age of the Best Actress winner is 50 years.
- The mean age for Best Actress recipients is 33.6 years.
- The standard deviation for Best Actress recipients is 12.7 years.
For the Best Actress:
[tex]\[ X = 50 \][/tex]
[tex]\[ \mu = 33.6 \][/tex]
[tex]\[ \sigma = 12.7 \][/tex]
Substituting these values into the formula:
[tex]\[ z_{\text{actress}} = \frac{50 - 33.6}{12.7} \approx 1.29 \][/tex]
So, the z-score for the Best Actress winner is [tex]\(1.29\)[/tex].
3. Comparing the z-scores to determine who had the more extreme age:
To determine who had the more extreme age, we compare the absolute values of the z-scores:
[tex]\[ |z_{\text{actor}}| = |-1.65| = 1.65 \][/tex]
[tex]\[ |z_{\text{actress}}| = |1.29| = 1.29 \][/tex]
Since [tex]\(1.65\)[/tex] is greater than [tex]\(1.29\)[/tex], the winner of the Best Actor award had an age more extreme relative to the normal age distribution for their category.
Thus, since the [tex]$z$[/tex] score for the winner of Best Actor is [tex]\( -1.65\)[/tex] and the [tex]$z$[/tex] score for the winner of Best Actress is [tex]\(1.29\)[/tex] , the winner of Best Actor had the more extreme age.
