Question

Men tend to have longer feet than women. So, if you find a really long footprint at the scene of a crime, then in the absence of any other evidence, you would probably conclude that the criminal was a man. And conversely, if you find a really short footprint at the scene of a crime, then (again in the absence of any other information) you would probably conclude that the criminal was a woman. Where should the cut-offvalue be for concluding that the criminal is a man or a woman? And what is the probability that you will make a mistake? Suppose that men’s foot lengths are normally distributed with mean 9.84 inches and standard deviation 1.57 inches, and women’s foot lengths are normally distributed with mean 7.48 inches and standard deviation 1.18 inches. A reasonable starting point to deciding on a cut-off-value is to split the difference: conclude a footprint belongs to a man if it is longer than 8.66 inches (the midpoint of the means 7.48 and 9.84).

1. Using this rule, what is the probability that you will mistakenly identify a man’s footprint as having come from a woman?

2. Using this rule again, what is the probability that you will mistakenly identify a woman’s footprint as having come from a man?

3. Change the cut-off-value (from 8.66 inches to some new value) so that the error probability from #1 is reduced to 0.08.

4. Determine the probability of mistakenly identifying a woman’s footprint as having come from a man using the new cut-off-value from #3.

5. Comment on how changing the cut-off-value affected the two kinds of error probabilities. Also explain why it makes sense that the probabilities changed as they did.

Answer 1