To determine the number of women who ran faster than Joan, we need to calculate the percentage of women who were slower than her and then apply that percentage to the total number of women in her age group.
Given that Joan's finishing time was 1.81 standard deviations faster than the women's average for her age group, we can use the properties of a normal distribution to find the corresponding percentage.
Since Joan is faster than the average, her finishing time would fall in the top portion of the distribution. Using a standard normal distribution table or a calculator, we can find the percentage of data below her finishing time. The Z-score associated with 1.81 standard deviations is approximately 0.9641, which corresponds to a percentage of 96.41%.
This means that approximately 96.41% of the women in her age group ran slower than Joan. To find the number of women who ran faster, we subtract this percentage from 100%: 100% - 96.41% = 3.59%.
To determine the number of women, we multiply the percentage by the total number of women in her age group: 3.59% * 410 = 14.709.
Rounding down to the nearest whole number, we can conclude that approximately 14 women ran faster than Joan.