Sure, let's solve this step-by-step.
1. Understand the Problem:
- We have a normally distributed variable, which in this case is the time taken by 15-year-olds to run a certain race.
- The mean (average time) is given as [tex]\(\mu = 18\)[/tex] seconds.
- The standard deviation is [tex]\(\sigma = 1.2\)[/tex] seconds.
- We want to find the percentage of runners who complete the race in less than [tex]\(14.4\)[/tex] seconds.
2. Calculate the Z-Score:
- The Z-score tells us how many standard deviations away a data point is from the mean.
- The formula for the Z-score [tex]\(z\)[/tex] is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where [tex]\(x\)[/tex] is the value we are interested in (14.4 seconds in this case).
Plugging in the values:
[tex]\[
z = \frac{14.4 - 18}{1.2} = \frac{-3.6}{1.2} = -3.0
\][/tex]
3. Find the Probability:
- The Z-score of [tex]\(-3.0\)[/tex] tells us how far 14.4 seconds is from the mean in terms of standard deviations.
- We use the cumulative distribution function (CDF) for a standard normal distribution to find the probability that a value is less than a given Z-score.
- The CDF value for a Z-score of [tex]\(-3.0\)[/tex] is approximately [tex]\(0.0013498980316300933\)[/tex].
This value represents the probability that a runner completes the race in less than [tex]\(14.4\)[/tex] seconds.
4. Convert the Probability to Percentage:
- To convert the probability to a percentage, we multiply by 100:
[tex]\[
\text{Percentage} = 0.0013498980316300933 \times 100 \approx 0.13498980316300932 \%
\][/tex]
5. Select the Appropriate Answer:
- Given the options:
[tex]\[
\begin{array}{c}
0.15 \% \\
0.30 \% \\
0.60 \% \\
2.50 \% \\
\end{array}
\][/tex]
- The closest and correct answer is [tex]\(0.15\%\)[/tex].
Therefore, the percentage of runners who have times less than [tex]\(14.4\)[/tex] seconds is approximately [tex]\(0.15\%\)[/tex].
