Question

The mean finish time for a yearly amateur auto race was 186.32 minutes with a standard deviation of 0.305 minute. The winning​ car, driven by Roger​, finished in minutes. The previous​ year's race had a mean finishing time of with a standard deviation of minute. The winning car that​ year, driven by ​, finished in minutes. Find their respective​ z-scores. Who had the more convincing​ victory? had a finish time with a​ z-score of nothing. had a finish time with a​ z-score of nothing.

Answer 1

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  $\mu_c = 186.32 \ minutes$

   The standard deviation is  $\sigma_c = 0.305 \ minutes$

   The time taken by  the winning car for the current year  is  $x = 185.85 \ minutes$

  The mean of the previous  year  is  $\mu_p = 110.7 \ minutes$

   The standard deviation the previous  year  is  $\sigma_p = 0.115 \ minutes$

 The time taken by  the winning car for the previous year  is  $y = 110.48 \ minutes$    

Generally the z-score for current year is mathematically evaluated as

     $z_c = \frac{x- \mu_c}{\sigma_c }$

=>     $z_c = \frac{ 185.85- 186.32}{0.305 }$

=> $z_c = -1.536$

Generally the z-score for previous year is mathematically evaluated as

     $z_p = \frac{y- \mu_p}{\sigma_p }$

=>     $z_p = \frac{ 110.48-110.7}{0.115 }$

=> $z_p = -1.913$

From the value obtained we see that  $z_p < z_c$ , Hence  Karen had a more convincing victory because of the lower z -score.