Question

the ages of the winners of a cycling tournament are approximately bell-shaped. the mean age is 28.7 years, with a standard deviation of 3.4 years. the winner in one recent year was 34 years old.

(a) transform the age to a z-score.
(b) interpret the results.
(c) determine whether the age is unusual.

(a) transform the age to a z-score.
\\(z = 1.56\\)
(type an integer or decimal rounded to two decimal places as needed.)

(b) interpret the results.
an age of 34 is standard deviation(s) the mean.
(type an integer or decimal rounded to two decimal places as needed.)

below
above

Explanation:

Calculate the z-score

Using the Z-Score Calculation knowledge point

$$ z = \frac{x - \mu}{\sigma} = \frac{34 - 28.7}{3.4} = \frac{5.3}{3.4} \approx 1.56 $$

Interpret the z-score

Using the Z-Score Interpretation knowledge point
An age of 34 is \(1.56\) standard deviations above the mean, because the z-score is positive.

Determine if the age is unusual

Using the Normal Distribution Outliers knowledge point
An observation is generally considered unusual if its z-score is less than \(-2\) or greater than \(2\). Since \(1.56\) is between \(-2\) and \(2\), the age of 34 is not unusual.

Answer:

Question a

\(z = 1.56\)

Question b

An age of 34 is 1.56 standard deviation(s) above the mean.

Question c

The age is not unusual because its z-score is between \(-2\) and \(2\).