Men tend to have longer feet than women So if you find a rea

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. If the converse was true, then you would probably conclude the criminal was a woman. But where should this cutoff be – and what is the probability you would make a mistake? Suppose men’s footprints are normally distributed with mean 25 cm and standard deviation 4 cm and women’s footprints are normally distributed with a mean of 19 cm and standard deviation 3 cm. A reasonable cut-off would be to conclude the footprint is a man’s if it is longer than 22 cm (halfway between the two means).

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

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

c.Notice that the probabilities for (a) and (b) are different. What would the cut-off value need to be in order for these probabilities to be equal? (And what is the probability?)

d. Now using your new value(s) from (c), if we found 4 different footprints at a crime-scene, what is the probability that we misclassify more than one of them?

Solution