document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=95603>Question 95603</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0536970378/algebrahomeworkh>Introductory Statistics</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\">   Recidivism. In May 2003 issue of Scientific American, R. Doyle examined rehabilitation of felons in the article, \"Reducing Crime: Rehabilitation is making a Comeback\". \r<BR>\n" );
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document.write( "  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. \r<BR>\n" );
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document.write( "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?\r<BR>\n" );
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document.write( "B. Determine the probability distribution of the random variable Y, the number of released prisoners of the six who are arrested  within 3 years.\r<BR>\n" );
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document.write( "C. Determine and interpret the mean of the random variable Y.\r<BR>\n" );
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document.write( "D. Obtain the standard deviation of Y.\r<BR>\n" );
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document.write( "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.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #69671 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=69671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 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.<BR>\n" );
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document.write( "Comment: This is a binary problem with n=6 and p =0.82<BR>\n" );
document.write( "Hopefully you have a TI calculator with Statistics functions.<BR>\n" );
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document.write( "A. Determine the probability that the number rearrested within 3 years will be exacly four<BR>\n" );
document.write( "P(x=4) = 6C4(0.82)^4 = binompdf(6,0,82,4)= 0.2197...<BR>\n" );
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document.write( "at least four<BR>\n" );
document.write( "P(4<=x<=6) = 1-binomcdf(6,0.82,3) = 0.9241....<BR>\n" );
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document.write( "at most five<BR>\n" );
document.write( "P(0<=x<=5) = binomcdf(6,0.82,5) = 0.69599...<BR>\n" );
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document.write( " between two and five; inclusive<BR>\n" );
document.write( "P(2<=x<=5) = binomcdf(6,0.82,5)-binomcdf(6.0.82,1) = 0.6950...<BR>\n" );
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document.write( "B. Determine the probability distribution of the random variable Y, the number of released prisoners of the six who are arrested within 3 years.<BR>\n" );
document.write( "Use binomcdf(6,0.82,Y) and keep changing the Y; let it range from 0 up to 6.<BR>\n" );
document.write( "The seven answers constitute the probability distribution.<BR>\n" );
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document.write( "C. Determine and interpret the mean of the random variable Y<BR>\n" );
document.write( "Use the fact that the mean for a binomial distribution<BR>\n" );
document.write( "is np. In you case that would be 6*0.82 = 4.92<BR>\n" );
document.write( "On the average close to 5 of 6 prisoners in the stated age category<BR>\n" );
document.write( "are arrested again within 3 years of being released.<BR>\n" );
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document.write( "D. Obtain the standard deviation of Y.<BR>\n" );
document.write( "standard deviation for a binomial distribution is sqrt(npq)<BR>\n" );
document.write( "= sqrt(6*0.82*0.18) = 0.9411<BR>\n" );
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document.write( "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.<BR>\n" );
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document.write( "I'm not sure if you are looking for a Hypothesis test here, or just an opinion<BR>\n" );
document.write( "based on the probabilities, or a judgement based on the normal curve.<BR>\n" );
document.write( "Anyway, here is the probilty of that happening if you assume that p = 0.82:<BR>\n" );
document.write( "P(x=2) = binompdf(6,0.82,2) = 0.0105... or about one out of 100 instead of<BR>\n" );
document.write( "82 out of 100.<BR>\n" );
document.write( "I'll leave the conjecturing to you.<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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