Question

Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 146 minutes with a standard deviation of 11 minutes. Consider 49 of the races. Let x = the average of the 49 races. Part (a) two decimal places.) Give the distribution of X. (Round your standard deviation to two decimal places)Part (b) Find the probability that the average of the sample will be between 144 and 149 minutes in these 49 marathons. (Round your answer to four decimal places.) Part (c) Find the 80th percentile for the average of these 49 marathons. (Round your answer to two decimal places.) __ min Part (d) Find the median of the average running times ___ min

Answer 1

The distribution of the sample mean for 49 world-class runners is normal with a mean of 146 minutes and a standard deviation of 1.57 minutes. The probability of the sample mean falling between 144 and 149 minutes, the 80th percentile, and the median can be calculated using the mean, standard deviation, and z-scores.

Step-by-step explanation:

Given a category of world-class runners known to run marathons in an average of 146 minutes with a standard deviation of 11 minutes, we have a sample size (n) of 49. Here is how to address each part of the question using statistical concepts:

Part a: Distribution of X

The distribution of the sample means will be approximately normal according to the Central Limit Theorem. The expected mean (mean of X) is the same as the population mean, which is 146 minutes. The standard deviation of the sample mean (standard deviation of X), also known as the standard error, is given by the formula σ_X = σ / √n. Hence, σ_X = 11 / √49 = 11 / 7 = 1.57 minutes, which is rounded to two decimal places.

Part b: Probability of Sample Mean Between 144 and 149 Minutes

To find the probability that the sample mean will be between 144 and 149 minutes, we calculate the z-scores for 144 and 149 and use the standard normal distribution to find this probability.

Part c: 80th Percentile of X

The 80th percentile for the average can be found by using z-tables to find the z-score corresponding to the 80th percentile and then converting this z-score into the actual time using the sample mean distribution characteristics (mean and standard deviation).

Part d: Median of Running Times

The median of a normally distributed variable is the same as its mean. Thus, the median running time is also 146 minutes.

Answer 2

The distribution of X, the average of the 49 races, follows a normal distribution with the same mean as the individual races and a standard deviation equal to the standard deviation of the individual races divided by the square root of the sample size. The probability that the average of the sample will be between 144 and 149 minutes can be found by standardizing the values and using the z-score formula. The 80th percentile for the average of these 49 marathons can be found by finding the corresponding z-score and using the formula for the corresponding value in the distribution. The median of the average running times is equal to the population mean.

Step-by-step explanation:

Part (a):

The distribution of X, the average of the 49 races, follows a normal distribution with the same mean as the individual races but with a standard deviation equal to the standard deviation of the individual races divided by the square root of the sample size. In this case, the standard deviation of X is 11 minutes / sqrt(49) = 1.57 minutes.

Part (b):

To find the probability that the average of the sample will be between 144 and 149 minutes, we need to standardize the values using the formula z = (x - μ) / σ, where z is the z-score, x is the value we want to find the probability for, μ is the population mean, and σ is the population standard deviation.

Part (c):

To find the 80th percentile for the average of these 49 marathons, we need to find the z-score that corresponds to the 80th percentile and then use the formula x = z * σ + μ to find the corresponding value in the distribution.

Part (d):

The median of the average running times is equal to the population mean, which is 146 minutes.