The ages of the winners of a cycling tournament are approximately​ bell-shaped. The mean age is 27.2 ​years, with a standard deviation of 3.3 years. The winner in one recent year was 20 years old.
​(a) Transform the age to a​ z-score.
​(b) Interpret the results.
​(c) Determine whether the age is unusual.

Let's solve this step-by-step.

### Given Data
- Mean age of winners, \( \mu = 27.2 \) years
- Standard deviation, \( \sigma = 3.3 \) years
- Age of the recent winner, \( X = 20 \) years

### Solution

#### (a) Transform the age to a z-score.

The formula for calculating a z-score is:

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

Substitute the values:

\[
z = \frac{20 - 27.2}{3.3}
\]
\[
z = \frac{-7.2}{3.3} \approx -2.18
\]

So, the z-score is approximately \( z = -2.18 \).

#### (b) Interpret the results.

The z-score of \(-2.18\) means that the age of 20 years is 2.18 standard deviations below the mean age of 27.2 years. In the context of a normal distribution, this indicates that the winner's age is notably younger than the average winner.

#### (c) Determine whether the age is unusual.

In a normal distribution, data points that fall more than 2 standard deviations away from the mean (either above or below) are generally considered unusual. Since this winner's age has a z-score of \(-2.18\), which is beyond \(-2\), the age of 20 years can be considered unusual among the winners of this tournament.

### Summary
- **z-score**: \(-2.18\)
- **Interpretation**: The winner is younger than the average by 2.18 standard deviations.
- **Unusual**: Yes, the age is considered unusual as it is more than 2 standard deviations from the mean.

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### Related Questions
1. What would be the z-score for a winner aged 30 years?
2. If the age distribution had a different standard deviation, how would that affect the z-score?
3. How can we use z-scores to determine whether other ages are unusual?
4. What does a positive z-score indicate in this context?
5. Can we conclude anything about the shape of the distribution just by knowing the mean and standard deviation?

### Tip
Using z-scores is a quick way to determine how far a data point lies from the mean, and whether it’s unusual based on typical cutoff values like ±2.

The ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 27.2 years, with a standard deviation of 3.3 years. The winner in one recent year was 20 years old. (a) Transform the age to a z-score. (b) Interpret the results. (c) Determine whether the age is unusual.

Determine the z-score for a cycling tournament winner aged 20 with a mean age of 27.2 years and a standard deviation of 3.3. Understand the interpretation of z-scores in a normal distribution and identify whether an age is considered unusual based on statistical principles.

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