Question

a race director is preparing for an upcoming marathon and estimates that the mean time to finish is 326 minutes. assume that the times are normally distributed, with a standard deviation of 50 minutes. use this table or the aleks calculator to find the percentage of times that are longer than 334 minutes. for your intermediate computations, use four or more decimal places. give your final answer to two decimal places (for example 98.23%).

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 334$, $\mu=326$ and $\sigma = 50$.
$z=\frac{334 - 326}{50}=\frac{8}{50}=0.16$

Step2: Find the cumulative probability for the z - score

Using the standard normal distribution table, the cumulative probability $P(Z\leq0.16)$ is approximately $0.5636$.

Step3: Find the probability of times longer than 334 minutes

We want $P(X>334)$, which is equivalent to $P(Z > 0.16)$ in the standard - normal distribution. Since $P(Z>z)=1 - P(Z\leq z)$, then $P(Z > 0.16)=1 - 0.5636 = 0.4364$.

Step4: Convert to percentage

To convert the probability to a percentage, we multiply by 100. So the percentage is $0.4364\times100 = 43.64\%$.

Answer:

$43.64\%$