Question

the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 27.4 years, with a standard deviation of 3.7 years. the winner in one recent year was 24 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 $x = 24$, $\mu=27.4$, and $\sigma = 3.7$.

Step3: Calculate z - score

$z=\frac{24 - 27.4}{3.7}=\frac{- 3.4}{3.7}\approx - 0.92$

Step4: Interpret the z - score

A z - score of approximately $-0.92$ means that the age of 24 years is approximately $0.92$ standard deviations below the mean age of the winners.

Step5: Determine if the age is unusual

In a normal distribution, values with a z - score outside the range of $- 2$ to $2$ are considered unusual. Since $-2<-0.92 < 2$, the age of 24 years is not unusual.

Answer:

(a) $z\approx - 0.92$
(b) The age of 24 years is approximately $0.92$ standard deviations below the mean age of the winners.
(c) The age is not unusual.