In a recent awards​ ceremony, the age of the winner for the Best Actor award was 

3737 

and the age of the winner for the Best Actress award was 

5151. 

For all recipients of Best​ Actor, the mean age is 

45.545.5 

years and the standard deviation is 

8.38.3 

years. For all recipients of Best​ Actress, the mean age is 

33.133.1 

years and the standard deviation is 

10.910.9 

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.

To determine who had the more "extreme" age when winning the award, we will use the **z-score**. The z-score tells us how many standard deviations an individual's age is from the mean in their respective category. The formula for the z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x \) is the individual’s age.
- \( \mu \) is the mean age of all winners in that category.
- \( \sigma \) is the standard deviation of ages for that category.

### Step 1: Calculate the z-score for the Best Actor winner
- Age of Best Actor winner: \( x = 37 \)
- Mean age for Best Actor: \( \mu = 45.5 \)
- Standard deviation for Best Actor: \( \sigma = 8.3 \)

The z-score for the Best Actor winner is:
\[
z_{\text{actor}} = \frac{37 - 45.5}{8.3} = \frac{-8.5}{8.3} \approx -1.02
\]

### Step 2: Calculate the z-score for the Best Actress winner
- Age of Best Actress winner: \( x = 51 \)
- Mean age for Best Actress: \( \mu = 33.1 \)
- Standard deviation for Best Actress: \( \sigma = 10.9 \)

The z-score for the Best Actress winner is:
\[
z_{\text{actress}} = \frac{51 - 33.1}{10.9} = \frac{17.9}{10.9} \approx 1.64
\]

### Step 3: Compare the z-scores
- The z-score for the Best Actor winner is approximately \(-1.02\), meaning the Best Actor winner was 1.02 standard deviations **below** the mean age.
- The z-score for the Best Actress winner is approximately \(1.64\), meaning the Best Actress winner was 1.64 standard deviations **above** the mean age.

### Conclusion:
The Best Actress winner had a higher absolute z-score (\(1.64\) compared to \(1.02\)), meaning she was further from the average age in her category than the Best Actor winner was in his category. Therefore, **the Best Actress winner had the more extreme age** relative to the average age in her category.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How is the z-score used in comparing data from different distributions?
2. Can z-scores be used to compare more than two groups?
3. What does a negative z-score represent in this context?
4. How would the results change if the standard deviations were different?
5. How could we interpret z-scores in other contexts, such as performance metrics?

**Tip:** When comparing different groups, using z-scores standardizes the data, allowing for comparison even if the groups have different means and standard deviations.

In a recent awards​ ceremony, the age of the winner for the Best Actor award was 37 and the age of the winner for the Best Actress award was 51. For all recipients of Best​ Actor, the mean age is 45.5 years and the standard deviation is 8.3 years. For all recipients of Best​ Actress, the mean age is 33.1 years and the standard deviation is 10.9 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.

Learn how to compare the ages of Best Actor and Best Actress winners using z-scores. This example covers key statistical concepts, including mean, standard deviation, and z-scores, with a detailed explanation of the more extreme age relative to the category's average. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation