Complete Question
The mean finish time for a yearly amateur auto race was 186.94 minutes with a standard deviation of 0.305 minute. The winning car, driven by Roger, finished in 185.85 minutes. The previous year's race had a mean finishing time of 110.7 with a standard deviation of 0.115 minute. The winning car that year, driven by Karen, finished in 110.48 minutes.
Find their respective z-scores.
Who had the more convincing victory?
A. Roger had a more convincing victory because of a higher z-score.
B. Karen a more convincing victory because of a higher z-score.
C. Roger had a more convincing victory, because of a lower z-score.
D. Karen a more convincing victory because of a lower z-score.
Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The mean of the current year is ![\mu_c = 186.32 \ minutes](https://tex.z-dn.net/?f=%5Cmu_c%20%3D%20%20186.32%20%5C%20%20minutes)
The standard deviation is ![\sigma_c = 0.305 \ minutes](https://tex.z-dn.net/?f=%5Csigma_c%20%20%3D%20%200.305%20%5C%20%20minutes)
The time taken by the winning car for the current year is ![x = 185.85 \ minutes](https://tex.z-dn.net/?f=x%20%3D%20%20185.85%20%5C%20minutes)
The mean of the previous year is ![\mu_p = 110.7 \ minutes](https://tex.z-dn.net/?f=%5Cmu_p%20%3D%20%20110.7%20%5C%20%20minutes)
The standard deviation the previous year is ![\sigma_p = 0.115 \ minutes](https://tex.z-dn.net/?f=%5Csigma_p%20%20%3D%200.115%20%20%5C%20%20minutes)
The time taken by the winning car for the previous year is
Generally the z-score for current year is mathematically evaluated as
![z_c = \frac{x- \mu_c}{\sigma_c }](https://tex.z-dn.net/?f=z_c%20%20%3D%20%20%5Cfrac%7Bx-%20%5Cmu_c%7D%7B%5Csigma_c%20%7D)
=>
=>
Generally the z-score for previous year is mathematically evaluated as
![z_p = \frac{y- \mu_p}{\sigma_p }](https://tex.z-dn.net/?f=z_p%20%20%3D%20%20%5Cfrac%7By-%20%5Cmu_p%7D%7B%5Csigma_p%20%7D)
=>
=>
From the value obtained we see that
, Hence Karen had a more convincing victory because of the lower z -score.