Final answer:
For a recent 10k run with a normal distribution of finishers, the percentage of finishers with times between 50 and 75 minutes is 82.28%. The percentage of finishers with times less than 80 minutes is 92.65%. The 35th percentile for the finishing times is approximately 56.24 minutes, and the seventh decile is approximately 68.77 minutes.
Step-by-step explanation:
a. To determine the percentage of finishers with times between 50 and 75 minutes, we need to find the area under the normal curve between these two values. First, we calculate the z-scores for 50 minutes and 75 minutes using the formula: z = (x - mean) / standard deviation. Using the mean of 62 minutes and standard deviation of 11 minutes, we find the z-score for 50 minutes is -1.18 and the z-score for 75 minutes is 1.18. We can then use a standard normal distribution table or a calculator to find the percentage of values between these z-scores, which is 0.8228 or 82.28%.
b. To find the percentage of finishers with times less than 80 minutes, we calculate the z-score for 80 minutes using the formula mentioned earlier. The z-score for 80 minutes is 1.45. We can then use a standard normal distribution table or a calculator to find the percentage of values less than this z-score, which is 0.9265 or 92.65%.
c. To obtain the 35th percentile for the finishing times, we need to find the corresponding z-score. We can use a standard normal distribution table or a calculator to find the z-score for a cumulative probability of 0.35, which is -0.3853. We can then use the formula mentioned earlier to find the corresponding finishing time: x = (z * standard deviation) + mean. Using the z-score of -0.3853, standard deviation of 11 minutes, and mean of 62 minutes, we find the 35th percentile for the finishing times is approximately 56.24 minutes.
d. To find the seventh decile for the finishing times, we need to find the z-score that corresponds to a cumulative probability of 0.7. Using a standard normal distribution table or a calculator, we find that the z-score is 0.5244. We can then use the formula mentioned earlier to find the corresponding finishing time: x = (z * standard deviation) + mean. Using the z-score of 0.5244, standard deviation of 11 minutes, and mean of 62 minutes, we find the seventh decile for the finishing times is approximately 68.77 minutes.