Respuesta :
Answer:
To be in the top 1% of runners, a man has to run the 400m in at most 55.8 seconds.
Step-by-step explanation:
Problems of normally distributed samples can be solved using 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.
In this problem, we have that:
[tex]\mu = 93, \sigma = 16[/tex]
How fast does a man have to run to be in the top 1% of runners (quickest runner)?
Quickest runner means that he ran in the least time, that is, his time is in the 1st percentile. So it is values of X and lower, in which X is found when Z has a pvalue of 0.01. So X when Z = -2.325.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-2.325 = \frac{X - 93}{16}[/tex]
[tex]X - 93 = -2.325*16[/tex]
[tex]X = 55.8[/tex]
To be in the top 1% of runners, a man has to run the 400m in at most 55.8 seconds.
