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Footprints Activity 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 c0nversely, 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: 1015 20 25 30 35 40 Centimeters 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 (ngrm_a_lgc_lf(-1 E99, 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 (ngrmalgd22, 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 Use (286m) and plot that value On your sketch. - Compute the probability of mistakenly identifying a footprint using ti84 calculator functions 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 longer, using the mean and standard deviation for women. Label this probability on your drawing. Include the calculator commands you used to cempute the probability. - Compute the probability of mistakenly identifying a footprint using ti84 calculator functions as a woman's using your foot length as the cut-off. In other words, find the probability of obtaining a footprint of your length or shorter, using the mean and standard deviation for men. Label this probability on your drawing. Include the calculator commands you used to cempute 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 get smaller? Explain why it makes sense that the probabilities changed as they did