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Sagot :
Using the normal distribution, it is found that:
a) The cutoff time for the fastest 5% of athletes in the men's group is of 3448.58.
b) The cutoff time for the slowest 10% of athletes in the women's group is of 6298.72.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, which is the percentile of X.
Item a:
For men, we have that [tex]\mu = 4378, \sigma = 565[/tex].
The cutoff for the fastest 5% of the athletes is the 5th percentile, which is X when Z has a p-value of 0.05, that is, X when Z = -1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 4378}{565}[/tex]
[tex]X - 4378 = -1.645(565)[/tex]
[tex]X = 3448.58[/tex]
The cutoff time for the fastest 5% of athletes in the men's group is of 3448.58.
Item b:
For women, we have that [tex]\mu = 5212, \sigma = 849[/tex].
The cutoff is the 90th percentile, which is X when Z = 1.28, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 5212}{849}[/tex]
[tex]X - 5212 = 1.28(849)[/tex]
[tex]X = 6298.72[/tex]
The cutoff time for the slowest 10% of athletes in the women's group is of 6298.72.
More can be learned about the normal distribution at https://brainly.com/question/24663213
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