Question

the ages of the winners of a cycling tournament are approximately bell - shaped. the mean age is 28.5 years, with a standard deviation of 3.4 years. the winner in one recent year was 30 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 30 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 outside of the range from - 2 to 2 is unusual. b. yes, this value is unusual. a z - score between - 2 and 2 is unusual. c. no, this value is not unusual. a z - score outside of the range from - 2 to 2 is not unusual. d. no, this value is not unusual. a z - score between - 2 and 2 is not unusual.

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation. Here, $x = 30$, $\mu=28.5$, and $\sigma = 3.4$.

Step2: Calculate the z - score

$z=\frac{30 - 28.5}{3.4}=\frac{1.5}{3.4}\approx0.44$

Step3: Interpret the z - score

A z - score of $z = 0.44$ means that an age of 30 is 0.44 standard deviations above the mean.

Step4: Determine if the value is unusual

In a normal distribution, values with z - scores outside the range of - 2 to 2 are considered unusual. Since $-2<0.44<2$, the age is not unusual.

Answer:

(a) $z = 0.44$
(b) An age of 30 is 0.44 standard deviation(s) above the mean.
(c) D. No, this value is not unusual. A z - score between - 2 and 2 is not unusual.