Question

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?

Answer 1

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.

Answer 2

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.