Final answer:
a) The distribution of X is Normal. b) The distribution of σ (standard deviation) is Binomial. c) The distribution of Στο Σα (sum of times) is Normal. d) The probability that a randomly selected runner's time will be between 21.8846 and 22.1846 minutes can be found using z-scores. e) The probability that the average time for the 41 runners is between 21.8846 and 22.1846 minutes can be calculated using the Central Limit Theorem. f) The probability that the randomly selected 41-person team will have a total time less than 906.1 minutes can be calculated using z-scores. g) The assumption of normal distribution is necessary for parts e) and f). h) The longest total time that a relay team can have and still make it to the championship round can be found by calculating the value corresponding to the top 15 percentile in the distribution of the total time.
Step-by-step explanation:
a. The distribution of X is A) Normal.
b. The distribution of σ (standard deviation) is C) Binomial.
c. The distribution of Στο Σα (sum of times) is A) Normal.
d. To find the probability that a randomly selected runner's time will be between 21.8846 and 22.1846 minutes, we can use the z-score formula. First, calculate the z-scores for the lower and upper bounds:
Lower Z-score: (21.8846 - 22) / 2.2 = -0.0521
Upper Z-score: (22.1846 - 22) / 2.2 = 0.0903
Using a standard normal distribution table (or a calculator), we can find the probabilities associated with these z-scores. The probability that the runner's time will be between 21.8846 and 22.1846 minutes is the difference between these two probabilities.
e. To find the probability that the average time for the 41 runners is between 21.8846 and 22.1846 minutes, we can use the Central Limit Theorem. The distribution of sample means will be approximately normal with a mean of the population mean (22 minutes) and a standard deviation of the population standard deviation divided by the square root of the sample size (2.2 / sqrt(41)). We can then calculate the z-scores for the lower and upper bounds and find the probabilities using a standard normal distribution table (or a calculator).
f. To find the probability that the randomly selected 41-person team will have a total time less than 906.1 minutes, we can calculate the z-score for this value using the formula: z = (X - μ) / σ. The mean (μ) for the total time of the team can be calculated by multiplying the mean time for an individual runner (22 minutes) by the total number of runners (41). The standard deviation (σ) for the total time can be calculated by multiplying the standard deviation for an individual runner (2.2 minutes) by the square root of the total number of runners. Once we have the z-score, we can find the probability associated with it.
g. For parts e) and f), the assumption of normal distribution is B) Yes, because we are using the Central Limit Theorem to approximate the distribution of the sample means and the total time.
h. The longest total time that a relay team can have and still make it to the championship round is determined by finding the value in the distribution of the total time that corresponds to the top 15 percentile. We can calculate the z-score for this percentile using a standard normal distribution table (or a calculator), and then use the formula z = (X - μ) / σ to find the corresponding time value.