Answered

Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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

We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.