62.3k views
3 votes
Running times for 400 meters are Normally distributed for young men between 18 and 30 years of age with a mean of 93 seconds and a standard deviation of 16 seconds. How fast does a man have to run to be in the top 1% of runners?

2 Answers

6 votes

Final answer:

To be in the top 1% of runners, a man would need to run the 400-meter dash faster than approximately 130.28 seconds, determined by using the z-score for the top 1% of a normal distribution and the given mean and standard deviation.

Step-by-step explanation:

To determine how fast a man must run to be in the top 1% of 400-meter runners, we must use the information given that the running times are normally distributed with a mean of 93 seconds and a standard deviation of 16 seconds. The first step is to find the z-score that corresponds to the top 1% of a normal distribution. This can typically be found in a z-table or by using a calculator with statistical functions. For a normal distribution, the z-score for the top 1% is approximately 2.33.

Next, we convert the z-score to the actual running time using the formula X = μ + (z × σ), where X is the running time we want to find, μ is the mean running time, z is the z-score, and σ is the standard deviation. Substituting the values we get: X = 93 + (2.33 × 16) which equals 130.28 seconds.

Therefore, a man would need to run faster than 130.28 seconds to be in the top 1% of 400-meter runners.

User Zoe Rowa
by
6.2k points
6 votes

Answer:

To be in the top 1% of the runners, the man has to run the 400 meters in at most 55.768 seconds.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
\mu and standard deviation
\sigma, the zscore of a measure X is given by:


Z = (X - \mu)/(\sigma)

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:


\mu = 93, \sigma = 16

How fast does a man have to run to be in the top 1% of runners?

The lower the time, the faster they are. So the man has to be at most in the 1st percentile, which is X when Z has a pvalue of 0.01. So he has to run in at most X seconds, and X is found when Z = -2.327. Then


Z = (X - \mu)/(\sigma)


-2.327 = (X - 93)/(16)


X - 93 = -2.327*16


X = 55.768

To be in the top 1% of the runners, the man has to run the 400 meters in at most 55.768 seconds.

User Will McCutchen
by
6.8k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.