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

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Understanding the Problem

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

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 seconds, and the standard deviation is 1.2 seconds.

Calculating the Z-Score

To find the percentage of runners who have times less than 14.4 seconds, 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=X−μσz = \frac{X - \mu}{\sigma}

where XX is the value we are interested in, μ\mu is the mean, and σ\sigma is the standard deviation.

Applying the Formula

We are interested in finding the percentage of runners who have times less than 14.4 seconds. So, we will use X=14.4X = 14.4, μ=18\mu = 18, and σ=1.2\sigma = 1.2 in the formula.

z=14.4−181.2=−3.61.2=−3z = \frac{14.4 - 18}{1.2} = \frac{-3.6}{1.2} = -3

Finding the Percentage

Now that we have the z-score, we can use a standard normal distribution table (also known as a z-table) to find the percentage of runners who have times less than 14.4 seconds. The z-table shows the area under the standard normal curve to the left of a given z-score.

Using the Z-Table

Looking up the z-score of -3 in the z-table, we find that the area to the left of -3 is approximately 0.0013. This means that 0.13% of the runners have times less than 14.4 seconds.

Conclusion

In conclusion, the percentage of runners who have times less than 14.4 seconds is approximately 0.13%.

Calculating the Percentage

To calculate the percentage, we can multiply the area to the left of the z-score by 100.

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

Final Answer

The final answer is: 0.13%\boxed{0.13\%}

Understanding the Problem

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

Q&A

Q: What is the normal distribution?

A: 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.

Q: What is the z-score?

A: 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=X−μσz = \frac{X - \mu}{\sigma}

where XX is the value we are interested in, μ\mu is the mean, and σ\sigma is the standard deviation.

Q: How do I calculate the z-score?

A: To calculate the z-score, you need to know the value you are interested in, the mean, and the standard deviation. You can use the formula:

z=X−μσz = \frac{X - \mu}{\sigma}

Q: What is the z-table?

A: The z-table is a table that shows the area under the standard normal curve to the left of a given z-score.

Q: How do I use the z-table?

A: To use the z-table, you need to look up the z-score in the table and find the area to the left of the z-score.

Q: What is the area to the left of a z-score?

A: The area to the left of a z-score is the percentage of the data that is less than the z-score.

Q: How do I calculate the percentage?

A: To calculate the percentage, you can multiply the area to the left of the z-score by 100.

Q: What is the final answer?

A: The final answer is 0.13%.

Common Misconceptions

Q: Is the normal distribution always symmetric?

A: No, the normal distribution is not always symmetric. However, in this case, the data is approximately normally distributed with a mean of 18 seconds and a standard deviation of σ=1.2\sigma = 1.2 seconds.

Q: Can I use the z-table for any z-score?

A: No, the z-table is only for z-scores between -3 and 3.

Q: How do I know if the data is normally distributed?

A: You can use a normality test to check if the data is normally distributed.

Conclusion

In conclusion, the percentage of runners who have times less than 14.4 seconds is approximately 0.13%. This is calculated by using the z-score formula, looking up the z-score in the z-table, and multiplying the area to the left of the z-score by 100.

Final Answer

The final answer is: 0.13%\boxed{0.13\%}