a runners distribution of times for running 1,000 meters has a mean of 4.5 minutes with a standard deviation…

Explanation:

Step1: Recall the z - score formula

The z - score formula is (z=\frac{x-\mu}{\sigma}), where (z) is the z - score, (x) is the data point, (\mu) is the mean, and (\sigma) is the standard deviation. We are given that (\mu = 4.5), (\sigma=0.75), and (z=- 1.76). We need to solve for (x).

Step2: Rearrange the z - score formula for (x)

Starting with (z=\frac{x - \mu}{\sigma}), we can multiply both sides by (\sigma) to get (z\sigma=x-\mu). Then add (\mu) to both sides: (x=\mu+z\sigma).

Step3: Substitute the given values into the formula

Substitute (\mu = 4.5), (z=-1.76), and (\sigma = 0.75) into (x=\mu+z\sigma). (x=4.5+(-1.76)\times0.75) First, calculate ((-1.76)\times0.75=-1.32). Then (x = 4.5-1.32). (x=3.18)

Answer:

3.18 minutes