the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 28.1 years…

the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 28.1 years, with a standard deviation of 3.5 years. the winner in one recent year was 25 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.)

the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 28.1 years, with a standard deviation of 3.5 years. the winner in one recent year was 25 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.)

Answer

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 $\mu = 28.1$, $\sigma=3.5$, and $x = 25$.

Step3: Calculate z - score

Substitute the values into the formula: $z=\frac{25 - 28.1}{3.5}=\frac{- 3.1}{3.5}\approx - 0.89$.

Step4: Interpret z - score

A z - score of approximately $-0.89$ means that the age of the winner ($25$ years old) is approximately $0.89$ standard deviations below the mean age of the winners ($28.1$ years old).

Step5: Determine if age is unusual

In a normal distribution, values with a z - score outside the range of $- 2$ to $2$ are considered unusual. Since $-2<-0.89 < 2$, the age of $25$ years old is not unusual.

Answer:

(a) $z\approx - 0.89$ (b) The age of the winner is approximately $0.89$ standard deviations below the mean age of the winners. (c) The age is not unusual.