Question

The distribution for triathlon time varies depending on the population you are describing. The distribution for men ages 30 - 34 is N(μ=4331, σ=597). The distribution for women ages 25 - 29 is N(μ=5286, σ=819). 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____.

Answer 1

a) The cutoff time for the fastest 5% of athletes in the men's group is approximately 3299.12 seconds. b) The cutoff time for the slowest 10% of athletes in the women's group is approximately 6414.72 seconds.

Step-by-step explanation:

a) To find the cutoff time for the fastest 5% of athletes in the men's group, we need to find the z-score corresponding to the 5th percentile. Since the distribution is normal, we can use the z-score formula: Z = (X - μ) / σ, where X is the cutoff time, μ is the mean, and σ is the standard deviation. Substituting the known values, we get Z = (X - 4331) / 597. Using a standard normal distribution table, we find that the z-score corresponding to the 5th percentile is approximately -1.645. We can rearrange the formula to solve for X: X = Z * σ + μ. Substituting the values, we get X = -1.645 * 597 + 4331 = 3299.115. Therefore, the cutoff time for the fastest 5% of athletes in the men's group is approximately 3299.12 seconds.

b) To find the cutoff time for the slowest 10% of athletes in the women's group, we follow the same steps as in part a. Using the z-score formula, we find that the z-score corresponding to the 90th percentile is approximately 1.28. Substituting the known values, we get X = 1.28 * 819 + 5286 = 6414.72. Therefore, the cutoff time for the slowest 10% of athletes in the women's group is approximately 6414.72 seconds.