Question

When investigators find a footprint at a crime scene they use some information about foot length in order to determine the criminal’s gender. If they find a very long footprint then they may conclude that the criminal is a man. But, without other evidence, how sure can the investigator be? Suppose that the distribution of men’s foot lengths (in centimeters) is approximately N (25, 4) and the distribution of women’s foot lengths is approximately N (19, 3). Please answer the following questions.

a)         Suppose that the investigator decides to use the midpoint of the means as a cutoff point; if a footprint is longer than 22 cm they will assume that the criminal is a man. How often will they make a mistake (i.e., what percentage of women have feet longer than 22cm and what percentage of men have feet shorter than 22 cm)?

b) Since it is not desirable to make a mistake so often, determine what the cutoff point needs to be (this will be a foot length) so that they will only misidentify a man’s footprint as a woman’s 5% of the time.

c) With this new cutoff point, what issue arises? (Hint: Take a look at how often a woman’s footprint will be misidentified as a man’s.)

Answer 1

a)

µ =    25      
σ =    4      
          
P( X ≤    22   ) = P( (X-µ)/σ ≤ (22-25) /4)  
=P(Z ≤   -0.750   ) =   0.2266

µ =    19                  
σ =    3                  
                      
P ( X ≥   22.00   ) = P( (X-µ)/σ ≥ (22-19) / 3)              
= P(Z ≥   1.000   ) = P( Z <   -1.000   ) =    0.1587  

Total = 0.2266+0.1587  

= 0.3853

b)

µ=   25                  
σ =    4                  
proportion=   0.05                  
                      
Z value at    0.05   =   -1.64   (excel formula =NORMSINV(   0.05   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   -1.64   *   4   +   25  
X   =   18.42   (answer)         

Cut off = 18.42

c)

µ =    19                  
σ =    3                  
                      
P ( X ≥   18.42   ) = P( (X-µ)/σ ≥ (18.4205854921941-19) / 3)              
= P(Z ≥   -0.193   ) = P( Z <   0.193   ) =    0.5766   (answer)

57.66% women's footpront might identify as Men

Thanks in advance!

revert back for doubt

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