Question

6.(13) Standard of Proof Recall that (when the null hypothesis is that an accused person is innocent) a Type I error occurs when an innocent person is found guilty, and that a Type II error occurs when a guilty person is found innocent. The expected social cost of judicial error in a case equals the (probability of a Type I error times the social cost of a Type 1 eror) + (probability of a Type II error times the social cost of a Type II error). Another social cost is the cost of collecting evidence. Let s denote the strictness of the standard of proof of guilt, which has a minimum value of zero and a maximum value of1. s-0 corresponds to "on the preponderance of the evidence". s-1 corresponds to judging a person to be guilty only if he is certainly guilty Table 1 below gives the assumed probabilities of Type I and Type II errors as a function of the strictness of the standard of proof, holding fixed the amount of evidence collected. The accused is in fact guilty with a probability of 0.5. For example, with s 0.2, the probability that the accused is innocent and is judged to be innocent is (0.5)(0.68) 0.34, the probability that the accused is innocent and is judged to be guilty is (0.5) (0.32) 0.16, so that the probability of a Type I error is 0.2, etc. s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 0,9 1.0 Prob 0.2 0.18 0.160.140.12 0.10 0.08 0.060.040.02 0.00 Type error Prob 0.2 0.21 0.210.240.24 0.29 0.29 0.360.360.450.5 Type a) (4) Fill in the blanks in Table 2 below, for the criminal case where the cost of a type I error is 20 and the cost of a Type II error is 4. 0.0 0.1 0.2 0.3 040.5 0.6 0.7 0.8 0.9 1.0 Type I expected cost Type II expected cost Total What therefore is the strictness of the standard of proof that minimizes the expected cost of judicial error (the "optimal" strictness of the standard of proof)?

Answer 1

a)4. In the given table we have the data of type I error and type II error, now in the following table we have to calculate the expected cost and we have also given type I error cost 20 and type II error cost 4.

s=

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Type I expected cost

0.2*20=4

0.18*20=3.6

0.16*20=3.2

0.14*20=2.8

0.12*20=2.4

0.1*20=2

0.08*20=1.6

0.06*20=1.6

0.04*20=0.8

0.02*20=0.4

0.00*20=0

Type II expected cost

0.2*4=0.8

0.21*4=0.84

0.22*4=0.88

0.23*4=0.92

0.24*4=0.96

0.24*4=0.96

0.29*4=1.16

0.36*4=1.44

0.36*4=1.44

0.45*4=1.80

0.5*4=2

Total expected cost

4+0.8=4.8

3.6+0.84=4.44

3.22+0.88=4.08

2.8+0.92=3.72

2.4+0.96=3.36

2+0.96=2.96

1.6+1.16=2.76

1.6+1.44=3.04

0.8+1.44=2.24

0.4+1.80=2.20

0+2=2

So the stricnesss of standard proof is the probability of type II error with 0.5 i.e when a guilty person is found to be innocent with probability 0.5. It means guilty person is innocent with probability half.