Questions: The ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 27.5 years, with a standard deviation of 3.7 years. The winner in one recent year was 23 years old. (a) Transform the age to a z-score. (b) Interpret the results. (c) Determine whether the age is unusual.

The ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 27.5 years, with a standard deviation of 3.7 years. The winner in one recent year was 23 years old.
(a) Transform the age to a z-score.
(b) Interpret the results.
(c) Determine whether the age is unusual.

Transcript text: The ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 27.5 years, with a standard deviation of 3.7 years. The winner in one recent year was 23 years old. (a) Transform the age to a $z$-score. (b) Interpret the results. (c) Determine whether the age is unusual.

Solution

Solution Steps

Step 1: Calculating the z-score

To calculate the z-score, we use the formula $z = \frac{X - \mu}{\sigma}$, where $X$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting the given values, $z = \frac{23 - 27.5}{3.7} = -1.22$.

Step 2: Interpreting the z-score

A z-score of -1.22 indicates that the data point is below the mean, specifically, 1.22 standard deviations below.

Step 3: Determining if the data point is unusual

The data point is not considered unusual as its z-score falls within the range of -2 to 2, which encompasses approximately 95% of the data in a normal distribution.

Final Answer:

The data point with a z-score of -1.22 is not unusual.

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