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 [tex]\mu_c = 186.32 \ minutes[/tex]
The standard deviation is [tex]\sigma_c = 0.305 \ minutes[/tex]
The time taken by the winning car for the current year is [tex]x = 185.85 \ minutes[/tex]
The mean of the previous year is [tex]\mu_p = 110.7 \ minutes[/tex]
The standard deviation the previous year is [tex]\sigma_p = 0.115 \ minutes[/tex]
The time taken by the winning car for the previous year is [tex]y = 110.48 \ minutes[/tex]
Generally the z-score for current year is mathematically evaluated as
[tex]z_c = \frac{x- \mu_c}{\sigma_c }[/tex]
=> [tex]z_c = \frac{ 185.85- 186.32}{0.305 }[/tex]
=> [tex]z_c = -1.536[/tex]
Generally the z-score for previous year is mathematically evaluated as
[tex]z_p = \frac{y- \mu_p}{\sigma_p }[/tex]
=> [tex]z_p = \frac{ 110.48-110.7}{0.115 }[/tex]
=> [tex]z_p = -1.913[/tex]
From the value obtained we see that [tex]z_p < z_c[/tex] , Hence Karen had a more convincing victory because of the lower z -score.
