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.

Question content area bottom

Part 1

​(a) Transform the age to a​ z-score.

z equalsz=2.182.18

​(Type an integer or decimal rounded to two decimal places as​ needed.)

Part 2

​(b) Interpret the results.

An age of

3636

is

enter your response here

standard​ deviation(s)

▼

the mean.

​(Type an integer or decimal rounded to two decimal places as​ needed

To solve this problem, let's go through each part step by step.

### Part 1: Transform the age to a z-score.

The formula for calculating a z-score is:

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

Where:
- \(X\) is the value (age of the winner)
- \(\mu\) is the mean
- \(\sigma\) is the standard deviation

Given:
- \(X = 36\)
- \(\mu = 28.6\)
- \(\sigma = 3.4\)

Now, substituting the values into the formula:

\[
z = \frac{(36 - 28.6)}{3.4} = \frac{7.4}{3.4} \approx 2.18
\]

So the z-score is approximately **2.18**.

### Part 2: Interpret the results.

The z-score of **2.18** means that the age of 36 years is **2.18 standard deviations** above the mean age of the winners.

### Part 3: Determine whether the age is unusual.

In general, a z-score greater than **2** or less than **-2** is considered unusual in a bell-shaped distribution. Since **2.18** is greater than **2**, the age of **36 years** can be considered **unusual**.

### Summary of Answers:

- **(a)** The z-score is **2.18**.
- **(b)** An age of **36** is **2.18 standard deviations above** the mean.
- **(c)** The age is **unusual** because it is more than **2 standard deviations** from the mean.

Would you like to dive deeper into any of these parts, or do you have any other questions? Here are some related questions to consider:

1. What does it mean for a distribution to be bell-shaped?
2. How can z-scores be used in other contexts?
3. What is the significance of the mean and standard deviation in a dataset?
4. How do you interpret negative z-scores?
5. Can z-scores help identify outliers in a dataset?

**Tip:** Always consider the context of the data when interpreting z-scores, as the implications can vary based on the field of study.

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 a cycling tournament winner's age of 36, given a mean age of 28.6 years and a standard deviation of 3.4. Interpret the z-score result and determine whether the age is statistically unusual. This problem explores z-scores, standard deviation, and normal distribution, ideal for students in Grades 10-12.

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