Question

Suppose that a category of world-class runners are known to run a marathon in an average of 147 minutes with a standard deviation of 12 minutes. Consider 49 of the races. Find the probability that the runner will average between 146 and 150 minutes in these 49 marathons. (Round your answer to two decimal places.)

Answer 1

Answer:

0.6524 = 65.24% probability that the runner will average between 146 and 150 minutes in these 49 marathons.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Average of 147 minutes with a standard deviation of 12 minutes.

This means that $\mu = 147, \sigma = 12$

Consider 49 of the races.

This means that $n = 49, s = \frac{12}{\sqrt{49}} = \frac{12}{7} = 1.7143$

Find the probability that the runner will average between 146 and 150 minutes in these 49 marathons.

This is the p-value of Z when X = 150 subtracted by the p-value of Z when X = 146. So

X = 150

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{150 - 147}{1.7143}$

$Z = 1.75$

$Z = 1.75$ has a p-value of 0.9599

X = 146

$Z = \frac{X - \mu}{s}$

$Z = \frac{146 - 147}{1.7143}$

$Z = -0.583$

$Z = -0.583$ has a p-value of 0.3075.

0.9599 - 0.3075 = 0.6524.

0.6524 = 65.24% probability that the runner will average between 146 and 150 minutes in these 49 marathons.