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.

| [tex]$z$[/tex] | Probability |
|-------|-------------|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 2.00 | 0.9772 |
| 3.00 | 0.9987 |

A. 16%
B. 32%
C. 34%
D. 84%



Answer :

To find the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes, let's go through the problem step-by-step:

1. Convert the mean and target times from hours and minutes to total minutes:
- Mean time: [tex]\(3\)[/tex] hours [tex]\(50\)[/tex] minutes
[tex]\[ 3 \text{ hours} \times 60 \text{ minutes/hour} + 50 \text{ minutes} = 230 \text{ minutes} \][/tex]
- Target time: [tex]\(3\)[/tex] hours [tex]\(20\)[/tex] minutes
[tex]\[ 3 \text{ hours} \times 60 \text{ minutes/hour} + 20 \text{ minutes} = 200 \text{ minutes} \][/tex]

2. Calculate the z-score for the target time:
The z-score formula is:
[tex]\[ z = \frac{\text{Target time} - \text{Mean time}}{\text{Standard Deviation}} \][/tex]
Substituting the given values:
[tex]\[ z = \frac{200 \text{ minutes} - 230 \text{ minutes}}{30 \text{ minutes}} = \frac{-30 \text{ minutes}}{30 \text{ minutes}} = -1.0 \][/tex]

3. Find the probability corresponding to the z-score:
Using the standard normal distribution table, we look up the probability for [tex]\(z = -1.0\)[/tex]. Since the standard normal table typically provides probabilities for positive z-values, we know that the symmetry of the normal distribution means:

For [tex]\( z = -1.0 \)[/tex], the probability is [tex]\(0.1587\)[/tex].

This means the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes is approximately [tex]\(15.87\%\)[/tex].

4. Select the correct choice:
Given the options: [tex]$16 \%$[/tex], [tex]$32 \%$[/tex], [tex]$34 \%$[/tex], [tex]$84 \%$[/tex], the closest to [tex]\(15.87\%\)[/tex] is:
[tex]\[ 16\% \][/tex]

Therefore, the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes is [tex]\(\boxed{16\%}\)[/tex].