Answer:
Terry had a finish time with a z-score of -1.45.
Jane had a finish time with a z-score of -2.12. Due to the lower z-score, Jane had the more convincing win.
Step-by-step explanation:
Z-score:
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question:
Whoever's z-score was lower, that is, took less time to complete the race compared to their competitors, had the more convincing win.
Terry:
The mean finish time for a yearly amateur auto race was 185.53 minutes with a standard deviation of 0.339 minute. The winning car, driven by Terry, finished in 185.04 minutes.
This means that ![\mu = 185.53, \sigma = 0.339, X = 185.04](https://tex.z-dn.net/?f=%5Cmu%20%3D%20185.53%2C%20%5Csigma%20%3D%200.339%2C%20X%20%3D%20185.04)
So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{185.04 - 185.53}{0.339}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B185.04%20-%20185.53%7D%7B0.339%7D)
![Z = -1.45](https://tex.z-dn.net/?f=Z%20%3D%20-1.45)
Terry had a finish time with a z-score of -1.45.
Jane:
The previous year's race had a mean finishing time of 111.6 with a standard deviation of 0.104 minute. The winning car that year, driven by Jane, finished in 111.38 minutes.
This means that ![\mu = 111.6, \sigma = 0.104, X = 111.38](https://tex.z-dn.net/?f=%5Cmu%20%3D%20111.6%2C%20%5Csigma%20%3D%200.104%2C%20X%20%3D%20111.38)
So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{111.38 - 111.6}{0.104}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B111.38%20-%20111.6%7D%7B0.104%7D)
![Z = -2.12](https://tex.z-dn.net/?f=Z%20%3D%20-2.12)
Jane had a finish time with a z-score of -2.12. Due to the lower z-score, Jane had the more convincing win.