Question

A manager (who happens to be a white Anglo) is responsible for hiring personnel in her department. She is accurate 95% of the time in assessing the qualifications of white Anglo candidates (that is, 95% of the candidates who are qualified she assesses as qualified and 95% of the candidates who are unqualified she assesses as unqualified). She is accurate only 80% of the time for candidates of color. She hires candidates (they then become employees) that she assesses as qualified. In the pool of candidates, 60% of white Anglo candidates and 60% of candidates of color are qualified.(1) what proportion of white Agnlo hirees are qualified?(2) what proportion of the person of color who are hired are qualified?

Answer 1

(a) The proportion of qualified anglo candidates hired is 96.6%.

(b) The proportion of qualified persons of color candidates hired is 85.7%.

Explanation:

For the anglo candidates, the manager will hire two groups:

Group 1: the ones that are qualified and she assess correctly

Group 2: the ones that are not qualified but she does not assess correctly

If we consider a group of 100 Anglo candidates, 60 are qualified. The probability that they are in the Group 1 is:

`P(G_1)=P(qualified=yes)*P(assess=right) = 0.6*0.95=0.57`

The probability of being in Group 2 is

`P(G_2)=P(qualified=no)*P(assess=wrong) =0.4*0.05= 0.02\\`

The proportion of qualified anglo candidates hired is equal to the probability of being in Group 1, over the sum of the probabilities of being in Group 1 and 2:

`P(qualified=yes;hired=yes)=(P(G_1))/(P(G_1)+P(G_2)) =(0.57)/(0.57+0.02)\\ \\P(qualified=yes;hired=yes)=(0.57)/(0.59)= 0.966`

The proportion of qualified anglo candidates hired is 96.6%.

For the people of color is the same analysis, but the probabilities of assessment are different.

Group 1: the ones that are qualified and she assess correctly

Group 2: the ones that are not qualified but she does not assess correctly

If we consider a group of 100 non-Anglo candidates, 60 are qualified. The probability that they are in the Group 1 is:

`P(G_1)=P(qualified=yes)*P(assess=right) = 0.6*0.8=0.48`

The probability of being in Group 2 is

`P(G_2)=P(qualified=no)*P(assess=wrong) =0.4*0.2= 0.08\\`

The proportion of qualified anglo candidates hired is equal to the probability of being in Group 1, over the sum of the probabilities of being in Group 1 and 2:

`P(qualified=yes;hired=yes)=(P(G_1))/(P(G_1)+P(G_2)) =(0.48)/(0.48+0.08)\\ \\P(qualified=yes;hired=yes)=(0.48)/(0.56)= 0.857`

The proportion of qualified persons of color candidates hired is 85.7%.