Question 95603This question is from textbook Introductory Statistics
: Recidivism. In May 2003 issue of Scientific American, R. Doyle examined rehabilitation of felons in the article, "Reducing Crime: Rehabilitation is making a Comeback".
One aspect of the article discussed recidivism of juvenile prisoners between 14 and 17 years old, indicating the 82% of those released in 1994 were rearrested within 3 years. Assuming that recidivism rate still applies today, solve the following problems for six newly released juvinelle prisoners between 14 and 17 years old.
A. Determine the probability that the number rearrested within 3 years will be exacly four; at least four;at most five; between two and five; inclusive?
B. Determine the probability distribution of the random variable Y, the number of released prisoners of the six who are arrested within 3 years.
C. Determine and interpret the mean of the random variable Y.
D. Obtain the standard deviation of Y.
E. If in fact, exactly two of the six released are rearrested within 3 years, would you be inclined to conclude that recidivism rate today has decreased from the 82% rate in 1994? Explain your reasoning.
This question is from textbook Introductory Statistics
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 82% of those released in 1994 were rearrested within 3 years. Assuming that recidivism rate still applies today, solve the following problems for six newly released juvinelle prisoners between 14 and 17 years old.
-------------
Comment: This is a binary problem with n=6 and p =0.82
Hopefully you have a TI calculator with Statistics functions.
------------------------------------------------
A. Determine the probability that the number rearrested within 3 years will be exacly four
P(x=4) = 6C4(0.82)^4 = binompdf(6,0,82,4)= 0.2197...
------------
at least four
P(4<=x<=6) = 1-binomcdf(6,0.82,3) = 0.9241....
--------------
at most five
P(0<=x<=5) = binomcdf(6,0.82,5) = 0.69599...
--------------
between two and five; inclusive
P(2<=x<=5) = binomcdf(6,0.82,5)-binomcdf(6.0.82,1) = 0.6950...
----------------------
B. Determine the probability distribution of the random variable Y, the number of released prisoners of the six who are arrested within 3 years.
Use binomcdf(6,0.82,Y) and keep changing the Y; let it range from 0 up to 6.
The seven answers constitute the probability distribution.
-------------------
C. Determine and interpret the mean of the random variable Y
Use the fact that the mean for a binomial distribution
is np. In you case that would be 6*0.82 = 4.92
On the average close to 5 of 6 prisoners in the stated age category
are arrested again within 3 years of being released.
--------------------.
D. Obtain the standard deviation of Y.
standard deviation for a binomial distribution is sqrt(npq)
= sqrt(6*0.82*0.18) = 0.9411
-------------------
E. If in fact, exactly two of the six released are rearrested within 3 years, would you be inclined to conclude that recidivism rate today has decreased from the 82% rate in 1994? Explain your reasoning.
--------
I'm not sure if you are looking for a Hypothesis test here, or just an opinion
based on the probabilities, or a judgement based on the normal curve.
Anyway, here is the probilty of that happening if you assume that p = 0.82:
P(x=2) = binompdf(6,0.82,2) = 0.0105... or about one out of 100 instead of
82 out of 100.
I'll leave the conjecturing to you.
================
Cheers,
Stan H.
|
|
|
