Answered

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3.6 Triathlon times, Part II: The distribution for triathlon time varies depending on the population you are describing. The distribution for men ages 30 - 34 is N(μ=4378, σ=565). The distribution for women ages 25 - 29 is N(μ=5212, σ=849). Note, these distributions list the triathlon times in seconds. Use this information to compute each of the following. Report your answer to 2 decimal places.
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.
b) The cutoff time for the slowest 10% of athletes in the women's group.

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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