Question

Joan’s finishing time for the Bolder Boulder 10K race was 1.84 standard deviations faster than the women’s average for her age group. There were 395 women who ran in her age group. Assuming a normal distribution, how many women ran faster than Joan?

Answer 1

13 women

Explanation:

The area in the tail of the normal distribution for Z < 1.84 can be used to find the number of faster finishers.

Tail area

If times are normally distributed, women whose finish times are lower than -1.84 standard deviations from the mean will have shorter (faster) times than Joan's. The area of that tail of the distribution will be given by the function ...

normalcdf(lower Z limit, upper Z limit) = normalcdf(-∞, -1.84)

In the attached, we have used the value -12 in place of -∞, because the tail area below Z=-12 is essentially zero (relative to the area of interest).

The total number of women in the entire distribution is 395, so the number below Z=-1.84 is ...

395×normalcdf(-12, -1.84) ≈ 13

13 women ran faster than Joan.