Question

the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 28.9 years, with a standard deviation of 3.4 years. the winner in one recent year was 26 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 = (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Identify values

We are given that $\mu = 28.9$, $\sigma=3.4$ and $x = 26$.

Step3: Calculate z - score

Substitute the values into the formula: $z=\frac{26 - 28.9}{3.4}=\frac{- 2.9}{3.4}\approx - 0.85$.

Step4: Interpret the z - score

A z - score of approximately $-0.85$ means that the age of 26 years is approximately $0.85$ standard deviations below the mean age of the winners.

Step5: Determine if the age is unusual

Unusual values are typically considered to be those with a z - score less than $- 2$ or greater than $2$. Since $-2<-0.85 < 2$, the age of 26 years is not unusual.

Answer:

(a) $z\approx - 0.85$
(b) The age of 26 years is approximately $0.85$ standard deviations below the mean age of the winners.
(c) The age is not unusual.