Men tend to have longer feet than women

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.

Men tend to have longer feet than women

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. What is the probability that you will make a mistake?

Suppose that men’s foot lengths are normally distributed with mean 25 centimeters and standard deviation 4 centimeters, and women’s foot lengths are normally distributed with mean 19 centimeters and standard deviation 3 centimeters. Here is a sketch of the two curves:

Men vs. Women
A reasonable cut-off value for determining if a footprint belongs to a man or a woman is to split the difference: conclude that a footprint belongs to a man if it is longer than 22 centimeters (the midpoint of the means 19 and 25). We can compute the probability of mistakenly identifying the gender of footprints using normal probabilities, which are shown in the drawing below.
The probability of mistakenly identifying a man’s footprint as having come from a woman is the probability of observing a 22 cm or shorter footprint using the normal distribution for men (normalcdf(-1E99, 22, 25, 4) = 0.2266, the blue area in the drawing).

The probability of mistakenly identifying a woman’s footprint as having come from a man is the probability of observing a
22 cm or longer footprint using the normal distribution for women (normal cdf (22, 1E99, 19, 3) = 0.1587, the red area in the drawing).

What if, instead of 22 cm, they used your foot length as the cut-off? Determine the probabilities of mistakenly identifying a footprint in that case.

Sketch the two normal curves on the same axis, and label them (the curves and the axis) clearly. Measure your foot length in centimeters and plot that value on your sketch.

Compute the probability of mistakenly identifying a footprint as a man’s using your foot length as the cut-off. In other words, find the probability of obtaining a footprint of your length or shorter. Use the mean and standard deviation for men. Label this probability on your drawing. Include the calculator commands you used to compute the probability.

Compute the probability of mistakenly identifying a footprint as a woman’s using your footlength as the cut-off. In other words, find the probability of obtaining a footprint of your length or longer. Use the mean and standard deviation for women. Label this probability on your drawing. Include the calculator commands you used to compute the probability.

Suppose you want to change the probability of mistakenly identifying a footprint. Choose whether you want to change the probability of misidentifying a man’s footprint as a woman’s or misidentifying a woman’s footprint as a man’s. Choose the probability you want and determine the cut-off value that would satisfy your new probability. Hint: You need to find a data value from a given percent. Include the calculator commands you used to compute the cut-off value.

Comment on how changing the cut-off value affected the two kinds of error probabilities. Which one got bigger and which one got smaller? Explain why it makes sense that the probabilities changed as they did.