Question

Within three years of release, about 67% of released prisoners were rearrested. Within five years of release, about 75.5% of released prisoners were rearrested. Use these recidivism rates to answer the following.

Step 1 of 4:

If we take a random sample of 75 former inmates three years after they were released. What is the probability less than half have been arrested since their release?

Step 2 of 4:

What would the probability be if we doubled our sample size?

Step 3 of 4:

If we consider a sample of size 75, what proportion would have to be rearrested within 5 years to be considered unusually high? [Express your answer as a decimal rounded to 4 places.]

Step 4 of 4:

If we examine 100 prisoners after 3 years, what recidivism rate would put the sample in the 34th percentile? [Express your answer as a decimal rounded to 4 places.]

Answer 1

Step 1:
p= 67% and n=75
E(Phat) = p =    67.00%  
V(Phat) = pq/n = 0.67*(1-0.67)/75 = 0.002948
SD(Phat) = sqrt(V(Phat)) = 0.0543

P( X<0.5) = ?
  
I know that, z = (X-mean)/(sd)  
z1 = (0.5-0.67)/0.0543)   -3.1308
hence, P( X<0.5) = P(Z<-3.1308) = NORMSDIST(-3.1308) = 0.0009

Step 2:
p=67% and n=150
E(Phat) = p =    67.00%  
V(Phat) = pq/n = 0.67*(1-0.67)/150 = 0.001474
SD(Phat) = sqrt(V(Phat)) = 0.0384

P( X<0.5) = ?
I know that, z = (X-mean)/(sd)  
z1 = (0.5-0.67)/0.0384) = -4.4271
hence, P( X<0.5) = P(Z<-4.4271) = NORMSDIST(-4.4271) = 0.000005
As the sample size is doubles, the probability halfed.

Step 3:
p= 75.5% and n=75
E(Phat) = p = 75.5%
V(Phat) = pq/n = 0.755*(1-0.755)/75 = 0.002466333
SD(Phat) = sqrt(V(Phat)) = 0.0497

Unusually high is more than mean+3*sd. that is more than 0.755+2*0.0497 = 0.8544
P( X>0.8544) = 1 - P(X<0.8544) = ?

I know that, z = (X-mean)/(sd) = (0.8544-0.755)/0.0497) = 2
hence, P( X>0.8544) = 1- P(Z<2) = 1 - NORMSDIST(2) = 0.0228

Step 4:
p=67% and n=100
E(Phat) = p = 67.00%  
V(Phat) = pq/n = 0.67*(1-0.67)/100 = 0.002211
SD(Phat) = sqrt(V(Phat)) = 0.0470  

P(Z<z) = 34%
z = NORMSINV(0.34)
z = -0.412463129
I know that, z = (X-mean)/sd  
Hence, (X-mean)/sd = -0.4125
X = -0.4125*0.047+0.67
X = 0.6506