Question

a runner’s distribution of times for running 1,000 meters has a mean of 4.5 minutes with a standard deviation of 0.75 minutes. one of the runner’s times has a z - score of - 1.75. what is the runner’s time?
options:
1.18 minutes
3.75 minutes
5.75 minutes
5.82 minutes

Explanation:

Step1: Recall z-score formula

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $z$ is the z-score, $x$ is the raw score (runner's time we need to find), $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

Step2: Rearrange formula to solve for $x$

Rearrange the z-score formula to isolate $x$: $x = \mu + z\sigma$

Step3: Substitute given values

Substitute $\mu = 4.5$, $z = -1.75$, $\sigma = 0.75$ into the formula:
$x = 4.5 + (-1.75)(0.75)$

Step4: Calculate the product first

Compute $(-1.75)(0.75) = -1.3125$

Step5: Compute final value of $x$

$x = 4.5 - 1.3125 = 3.1875 \approx 3.18$

Answer:

3.18 minutes