Investigation 4: Foot sizes (due Thur, Jan 19)

You may work with one other person on this assignment, handing in one report with both names.  Word-processed reports are preferred to hand-written ones.  You may use a normal table or Minitab or a Java applet to perform calculations involving probability distributions, but be sure to specify what you asked the computer to calculate when you provide the answer.

Men have longer feet than women do. So, if you find a really long footprint at the scene of a crime, then in the absence of any other information, you would probably conclude that the criminal was a man. And 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 you put the cut-off value for concluding that the criminal is a man or a woman? And what are the probabilities that you make a mistake?

Suppose that you learn that American adult men’s foot lengths are normally distributed with mean 27 centimeters and standard deviation 2 centimeters, and American adult women’s foot lengths are normally distributed with mean 20 centimeters and standard deviation 3 centimeters.

a) Draw a well-labeled sketch of these two normal curves on the same scale. [It’s fine to draw these by hand, or you can copy the picture from the applet by clicking the “print screen” button on your keyboard and then pasting the picture into Word.  If you do this, please use the “crop” tool to select only the relevant parts of the screen to submit.]

b) What proportion of adult American males have a foot size of at least 30 centimeters?

c) What proportion of adult American females have a foot size of at least 30 centimeters?

Suppose that you choose the cut-off value to be 23.5 centimeters (the midpoint of the male mean of 27 centimeters and the female mean of 20 centimeters). In other words, if you find a foot length of more than 23.5 centimeters, then you will predict that it came from a male. And if you find a foot length of 23.5 centimeters or less, then you will predict that it came from a female.

d) With this cut-off value of 23.5, what is the probability that a male will produce a footprint that leads you to (incorrectly) predict him to be a female?

e) With this cut-off value of 23.5, what is the probability that a female will produce a footprint that leads you to (incorrectly) predict her to be a male?

f) Suppose that you want to alter the cut-off value (i.e., change it from 23.5 to some other value) so that you reduce the probability of incorrectly predicting a male to be female (as in d) to be only .02. Determine what this new cut-off value should be.

g) With this new cut-off value from f), what is the probability of incorrectly predicting a female to be male (as in e).

h) How does the probability in g) compare to that in e)?  Explain why this makes sense.