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Use z-scores to compare the given values.

In a recent awards ceremony, the age of the winner for the Best Actor award was 38, and the age of the winner for the Best Actress award was 50.

For all recipients of Best Actor, the mean age is 46.4 years, and the standard deviation is 5.1 years.

For all recipients of Best Actress, the mean age is 33.6 years, and the standard deviation is 12.7 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 [tex]$z$[/tex] score for the winner of Best Actor is [tex]$z = \square$[/tex] and the [tex]$z$[/tex] score for the winner of Best Actress is [tex]$z = \square$[/tex], the winner of [tex]$\square$[/tex] had the more extreme age.

(Round to two decimal places.)

Sagot :

GinnyAnswer
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.
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