Question

1. the mean 100 - meter dash time for track athletes is 12.8 seconds (sd = 0.3).

a. find and interpret the z - score of an athlete running 12.2 seconds.
b. a non - track athlete has a mean of 13.9 seconds (sd = 0.5) and runs 13.1 seconds.
c. who performed better relative to their group? justify your answer using z - scores.

Explanation:

Part (a)

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the individual value, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Here, $x = 12.2$ seconds, $\mu=12.8$ seconds, and $\sigma = 0.3$ seconds.

Step2: Calculate the z - score

Substitute the values into the formula: $z=\frac{12.2 - 12.8}{0.3}=\frac{- 0.6}{0.3}=- 2$

Step3: Interpret the z - score

A z - score of - 2 means that the athlete's 100 - meter dash time of 12.2 seconds is 2 standard deviations below the mean time of track athletes.

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the individual value, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Here, $x = 13.1$ seconds, $\mu = 13.9$ seconds, and $\sigma=0.5$ seconds.

Step2: Calculate the z - score

Substitute the values into the formula: $z=\frac{13.1 - 13.9}{0.5}=\frac{-0.8}{0.5}=- 1.6$

To determine who performed better relative to their group, we compare the z - scores. A lower (more negative) z - score for a time - based event (since a lower time is better in a dash) means a better performance relative to the group.
The track athlete has a z - score of - 2, and the non - track athlete has a z - score of - 1.6. Since - 2<-1.6, the track athlete's time is further below their group's mean (in terms of standard deviations) than the non - track athlete's time is below their group's mean. In the context of a 100 - meter dash, a lower time is better, so the track athlete performed better relative to their group.

Answer:

The z - score is - 2. It means the athlete's time is 2 standard deviations below the mean of track athletes' times.

Part (b) (Assuming we need to find the z - score for the non - track athlete)