Answered

Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly 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 appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.