Respuesta :
Answer:
a) 3354 seconds
b) 6294 seconds
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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.
(a) The cutoff time for the fastest 5% of athletes in the men's group, i.e. those who took the shortest 5% of time to finish.
[tex]\mu = 4313, \sigma = 583[/tex]
The cutoff for the shortest 5% is the 5th percentile, which is X when Z has a pvalue of 0.05. So X when Z = -1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 4313}{583}[/tex]
[tex]X - 4313 = -1.645*583[/tex]
[tex]X = 3354[/tex]
Cutoff of 3354 seconds.
(b) The cutoff time for the slowest 10% of athletes in the women's group.
[tex]\mu = 5261, \sigma = 807[/tex]
The slowest 10% is the 10% that takes more time, so the cutoff is the 100 - 10 = 90th percentile, which is X when Z has a pvalue of 0.9. So X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 5261}{807}[/tex]
[tex]X - 5261 = 1.28*807[/tex]
[tex]X = 6294[/tex]
Cutoff time of 6294 seconds
