The Times Of All 15-year-olds Who Run A Certain Race Are Approximately Normally Distributed With A Mean Μ = 18 \mu = 18 Μ = 18 Sec And A Standard Deviation Σ = 1.2 \sigma = 1.2 Σ = 1.2 Sec. What Percentage Of The Runners Have Times Less Than 14.4 Sec?A.

Understanding the Problem

The problem presents a scenario where the times of all 15-year-olds who run a certain race are approximately normally distributed with a mean $\mu = 18$ sec and a standard deviation $\sigma = 1.2$ sec. We are asked to find the percentage of runners who have times less than 14.4 sec.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this case, the mean time is 18 sec, and the standard deviation is 1.2 sec.

Calculating the Z-Score

To find the percentage of runners who have times less than 14.4 sec, we need to calculate the z-score. The z-score is a measure of how many standard deviations an element is from the mean. The formula for the z-score is:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value we are interested in, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Applying the Z-Score Formula

We can now apply the z-score formula to find the z-score for a time of 14.4 sec.

$$z = \frac{14.4 - 18}{1.2} = \frac{-3.6}{1.2} = -3$$

Using a Z-Table or Calculator

To find the percentage of runners who have times less than 14.4 sec, we need to use a z-table or calculator. A z-table is a table that shows the probability of a value being less than a certain z-score. We can look up the z-score of -3 in a z-table to find the corresponding probability.

Finding the Probability

Using a z-table or calculator, we find that the probability of a value being less than -3 is approximately 0.0013. This means that about 0.13% of the runners have times less than 14.4 sec.

Conclusion

In conclusion, we have found that about 0.13% of the runners have times less than 14.4 sec. This is a very small percentage, indicating that it is extremely unlikely for a runner to have a time less than 14.4 sec.

Calculating the Percentage

To calculate the percentage, we can multiply the probability by 100.

$$\text{Percentage} = 0.0013 \times 100 = 0.13\%$$

Final Answer

The final answer is 0.13%.

Understanding the Problem

The problem presents a scenario where the times of all 15-year-olds who run a certain race are approximately normally distributed with a mean $\mu = 18$ sec and a standard deviation $\sigma = 1.2$ sec. We are asked to find the percentage of runners who have times less than 14.4 sec.

Q&A

Q: What is the mean time of the runners?

A: The mean time of the runners is 18 sec.

Q: What is the standard deviation of the times?

A: The standard deviation of the times is 1.2 sec.

Q: How do we calculate the z-score?

A: We calculate the z-score using the formula: $z = \frac{X - \mu}{\sigma}$, where $X$ is the value we are interested in, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Q: What is the z-score for a time of 14.4 sec?

A: The z-score for a time of 14.4 sec is -3.

Q: How do we find the percentage of runners who have times less than 14.4 sec?

A: We use a z-table or calculator to find the probability of a value being less than a certain z-score. In this case, the probability of a value being less than -3 is approximately 0.0013.

Q: What is the percentage of runners who have times less than 14.4 sec?

A: The percentage of runners who have times less than 14.4 sec is approximately 0.13%.

Q: Why is it unlikely for a runner to have a time less than 14.4 sec?

A: It is unlikely for a runner to have a time less than 14.4 sec because the probability of a value being less than -3 is very small, approximately 0.0013.

Q: How do we calculate the percentage?

A: We calculate the percentage by multiplying the probability by 100.

Q: What is the final answer?

A: The final answer is 0.13%.

Additional Questions

Q: What is the normal distribution?

A: The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Q: What is the z-score formula?

A: The z-score formula is: $z = \frac{X - \mu}{\sigma}$, where $X$ is the value we are interested in, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Q: How do we use a z-table or calculator?

A: We use a z-table or calculator to find the probability of a value being less than a certain z-score.

Conclusion

In conclusion, we have answered the questions related to the problem of finding the percentage of runners who have times less than 14.4 sec. We have used the z-score formula, a z-table or calculator, and the normal distribution to find the answer.

Final Answer

The final answer is 0.13%.