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

options: 16%, 32%, 34%, 84%

Explanation:

Step1: Convert time to minutes

Mean time: 3 hours 50 minutes = \(3\times60 + 50 = 230\) minutes.
Target time: 3 hours 20 minutes = \(3\times60 + 20 = 200\) minutes.

Step2: Calculate z - score

The formula for z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 200\), \(\mu=230\), \(\sigma = 30\).
So \(z=\frac{200 - 230}{30}=\frac{- 30}{30}=- 1\).

Step3: Use the standard normal table

The standard normal table gives the probability \(P(Z\leq z)\). We know that for a normal distribution, the total area under the curve is 1, and the distribution is symmetric about \(z = 0\).
We know that \(P(Z\leq0)=0.5\) and \(P(Z\leq1) = 0.8413\). By symmetry, \(P(Z\leq - 1)=1 - P(Z\leq1)=1 - 0.8413 = 0.1587\approx16\%\)

Answer:

16%