Question

The mean finish time for a yearly amateur auto race was 186.88 minutes with a standard deviation of 0.387 minutes. The winning car, driven by Frank, finished in 186.07 minutes. The previous year's race had a mean finishing time of 111.5 with a standard deviation of 0.107. The winning car that year, driven by Terry, finished in 111.24 minutes. Find the z-scores. Who had the more convincing victory?

Answer 1

Using the formula for z-scores, Frank's z-score is approximately -2.10, and Terry's z-score is approximately -2.43. Thus, Terry had the more convincing victory since his z-score is further from zero.

Step-by-step explanation:

To calculate the z-scores for Frank and Terry's winning times in their respective races, we use the formula z = (X - μ) / σ where X is the value in question, μ is the mean, and σ is the standard deviation.

For Frank, with a mean time of 186.88 minutes and a standard deviation of 0.387, his z-score is:
z = (186.07 - 186.88) / 0.387 = -0.811 / 0.387 ≈ -2.10.

For Terry, with a mean time of 111.5 minutes and a standard deviation of 0.107, his z-score is:
z = (111.24 - 111.5) / 0.107 = -0.26 / 0.107 ≈ -2.43.

Comparing the z-scores, we find that Terry's victory is more convincing because his z-score is further from zero, indicating that his finishing time is farther from the mean relative to the variability of the race times.