Question

When John runs the 400 meter dash, his finishing times are normally distributed with a mean of 71 seconds and a standard deviation of 2.5 seconds. If John were to run 33 practice trials of the 400 meter dash, how many of those trials would be slower than 67 seconds, to the nearest whole number?

Answer 1

2

Explanation:

You want to know the number of trials out of 33 that would be slower than 67 seconds when times are normally distributed with a mean of 71 and a standard deviation of 2.5 seconds.

CDF

The normal CDF is the integral of the normal probability distribution function from -∞ to the x-value of interest. Calculators and spreadsheets can scale the CDF to any appropriate value of mean (µ) and standard deviation (σ) you may choose.

Application

The value obtained for the normal CDF using X=67, µ=71, and σ=2.5 is about 0.0547993. This is the probability that an individual trial will be slower than 67 seconds. Multiplying that by 33 will give the expected number of trials out of 33 that are that slow.

33 · 0.0547993 ≈ 1.808 ≈ 2

It is expected that 2 of 33 practice trials will be slower than 67 seconds.

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Additional comment

Many tools are available for finding the relevant probability. The one shown in the second attachment provides a nice visual, but sometimes has errors in the probability beyond the 5th decimal place.