Question

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. 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. Find their respective​ z-scores. Who had the more convincing​ victory? Terry had a finish time with a​ z-score of nothing. Jane had a finish time with a​ z-score of nothing. ​(Round to two decimal places as​ needed.)

Answer 1

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.

Explanation:

Z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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`

So

`Z = (X - \mu)/(\sigma)`

`Z = (185.04 - 185.53)/(0.339)`

`Z = -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`

So

`Z = (X - \mu)/(\sigma)`

`Z = (111.38 - 111.6)/(0.104)`

`Z = -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.