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 (quickest runner)?

Answer 1

A man must run 400 meters in approximately 130.28 seconds or faster to be in the top 1% of quickest runners, based on a normal distribution with a mean of 93 seconds and a standard deviation of 16 seconds.

Step-by-step explanation:

To find how fast a man has to run to be in the top 1% of runners, we need to determine the z-score that corresponds to the top 1% in a normal distribution. The mean running time for the 400 meters is given as 93 seconds, with a standard deviation of 16 seconds.

First, we consult a standard normal distribution table, use a z-score calculator, or statistical software to find the z-score that corresponds to the top 1%. This value is approximately z = 2.33, which means that a runner must perform 2.33 standard deviations better than the mean to be in the top 1%.

To calculate the exact time, we use the formula:
X = μ + z × σ
Where X is the time the runner must achieve, μ is the mean time, z is the z-score, and σ is the standard deviation. Substituting the values, we get:
X = 93 + 2.33 × 16

Now, we perform the arithmetic:
X = 93 + 2.33 × 16
X = 93 + 37.28
X = 130.28 seconds

Therefore, a man must run 400 meters in approximately 130.28 seconds or faster to be among the top 1% of quickest runners.

Answer 2

To be in the top 1% of runners, a man has to run the 400m in at most 55.8 seconds.

Step-by-step explanation:

Problems of normally distributed samples can be solved using 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 problem, we have that:

`\mu = 93, \sigma = 16`

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

Quickest runner means that he ran in the least time, that is, his time is in the 1st percentile. So it is values of X and lower, in which X is found when Z has a pvalue of 0.01. So X when Z = -2.325.

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

`-2.325 = (X - 93)/(16)`

`X - 93 = -2.325*16`

`X = 55.8`

To be in the top 1% of runners, a man has to run the 400m in at most 55.8 seconds.