Question

the times of the runners in a marathon are normally distributed, with a mean of 3 hours and 50 minutes and a standard deviation of 30 minutes. what is the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes? use the portion of the standard normal table below to help answer the question

z	probability
0.50	0.6915
1.00	0.8413
2.00	0.9772
3.00	0.9987

16%
32%
34%
84%

Explanation:

Step1: Convert times to minutes

The mean $\mu = 3\times60 + 50=230$ minutes. The value $x = 3\times60+20 = 200$ minutes and the standard - deviation $\sigma = 30$ minutes.

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$. Substitute $x = 200$, $\mu = 230$ and $\sigma = 30$ into the formula: $z=\frac{200 - 230}{30}=\frac{- 30}{30}=-1$.

Step3: Use the standard normal table

Since the standard normal table gives probabilities for non - negative z - scores and the normal distribution is symmetric, $P(Z\leq - 1)=1 - P(Z\leq1)$. From the table, $P(Z\leq1)=0.8413$, so $P(Z\leq - 1)=1 - 0.8413 = 0.1587\approx16\%$.

Answer:

16%