The ages of the winners of a cycling tournament are approximately​ bell-shaped. The mean age is 28.628.6 ​years, with a standard deviation of 3.43.4 years. The winner in one recent year was 3636 years old.
​(a) Transform the age to a​ z-score.
​(b) Interpret the results.
​(c) Determine whether the age is unusual.


Given:
- The mean age, \( \mu = 28.6 \) years
- The standard deviation, \( \sigma = 3.4 \) years
- The age of a recent winner, \( x = 36 \) years

We can solve this problem step-by-step.

### Part (a): Transform the Age to a Z-Score

The formula to calculate a \( z \)-score is:
\[
z = \frac{x - \mu}{\sigma}
\]
Substituting the values:
\[
z = \frac{36 - 28.6}{3.4} = \frac{7.4}{3.4} \approx 2.18
\]

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

### Part (b): Interpret the Results

A \( z \)-score of 2.18 means that the winner's age of 36 years is **2.18 standard deviations above the mean age** of 28.6 years. In a normal distribution, most data points (about 95%) lie within 2 standard deviations of the mean. A \( z \)-score of 2.18 suggests that this age is above average but not extremely rare.

### Part (c): Determine Whether the Age is Unusual

In general, \( z \)-scores above 2 or below -2 are considered somewhat unusual, and those above 3 or below -3 are considered very unusual. Here, the \( z \)-score is 2.18, which is slightly above the threshold of 2, indicating that while the age is somewhat unusual, it is not extremely rare.

### Summary:
- **Z-score**: 2.18
- **Interpretation**: The age is above average by 2.18 standard deviations.
- **Unusualness**: The age of 36 is somewhat unusual but not extremely rare.

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

Here are 5 related questions for practice:
1. What is a \( z \)-score and why is it useful?
2. How would the \( z \)-score change if the winner was 30 years old?
3. In a normal distribution, what percent of values lie within one standard deviation of the mean?
4. What \( z \)-score indicates an extremely unusual value in a normal distribution?
5. How can \( z \)-scores help in comparing values from different distributions?

**Tip:** When interpreting a \( z \)-score, remember that positive scores indicate values above the mean, while negative scores indicate values below the mean.

The ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 28.6 years, with a standard deviation of 3.4 years. The winner in one recent year was 36 years old. (a) Transform the age to a z-score. (b) Interpret the results. (c) Determine whether the age is unusual.

Learn how to calculate the z-score for an age of 36 years given a mean age of 28.6 and standard deviation of 3.4 in a bell-shaped distribution. This example interprets the z-score result and assesses if the age is unusual, ideal for students in Grades 10-12 studying statistics.

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