Question

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 3.3 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.)
(b) interpret the results.
an age of 24 is standard deviation(s) the mean.
(type an integer or decimal rounded to two decimal places as needed.)
(c) determine whether the age is unusual. choose the correct answer below
a. yes, this value is unusual. a z - score between - 2 and 2 is unusual.
b. yes, this value is unusual. a z - score outside of the range from - 2 to 2 is unusual.
c. no, this value is not unusual. a z - score between - 2 and 2 is not unusual.
d. no, this value is not unusual. a z - score outside of the range from - 2 to 2 is not unusual.

Explanation:

Step1: Calculate z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 24$ (the value), $\mu=28.6$ (the mean), and $\sigma = 3.3$ (the standard deviation).
Substitute the values into the formula: $z=\frac{24 - 28.6}{3.3}=\frac{- 4.6}{3.3}\approx - 1.39$

Step2: Interpret the z - score

The z - score of - 1.39 means that an age of 24 is 1.39 standard deviations below the mean.

Step3: Determine if the value is unusual

A value is considered unusual if its z - score is less than - 2 or greater than 2. Since $-2<-1.39 < 2$, the value is not unusual. So the correct option is C.

Answer:

(a) $z\approx - 1.39$
(b) An age of 24 is 1.39 standard deviation(s) below the mean.
(c) C. No, this value is not unusual. A z - score between - 2 and 2 is not unusual.