document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019805>Question 1019805</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. A small town has 5600 residents. The residents in the town were asked whether or not they favored building a new bridge across the river. You are given the following information on the residents' responses, broken down by gender:\r<BR>\n" );
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document.write( "                                        Men          Women          Total\r<BR>\n" );
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document.write( "          In Favor                  1400            280               1680\r<BR>\n" );
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document.write( "          Opposed                  840           3080               3920\r<BR>\n" );
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document.write( "                Total                2240           3360               5600\r<BR>\n" );
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document.write( "A) What is the probability of a randomly selected resident being in opposed to the bridge?\r<BR>\n" );
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document.write( "B) What is the probability that a randomly selected resident is a man and is Opposed to the bridge?\r<BR>\n" );
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document.write( "C) What is the probability of a randomly selected resident being a man or Opposed to the bridge?\r<BR>\n" );
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document.write( "D) If a randomly selected resident is a man, what is the probability that he is in favor of the bridge?\r<BR>\n" );
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document.write( "E) Are gender and opinion about the bridge mutually exclusive events? Why?\r<BR>\n" );
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document.write( "F) Are gender and opinion about the bridge independent events? Why? Show some \"proof\" </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635788 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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/cgi-bin/show-face.mpl?user=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=635788\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "1. A small town has 5600 residents. The residents in the town were asked whether or not they favored building a new bridge across the river. You are given the following information on the residents' responses, broken down by gender:<BR>\n" );
document.write( "                                    Men          Women          Total<BR>\n" );
document.write( "          In Favor                  1400            280               1680<BR>\n" );
document.write( "          Opposed                  840           3080               3920<BR>\n" );
document.write( "                Total                2240           3360               5600<BR>\n" );
document.write( "A) What is the probability of a randomly selected resident being in opposed to the bridge?<BR>\n" );
document.write( "B) What is the probability that a randomly selected resident is a man and is Opposed to the bridge?<BR>\n" );
document.write( "C) What is the probability of a randomly selected resident being a man or Opposed to the bridge?<BR>\n" );
document.write( "D) If a randomly selected resident is a man, what is the probability that he is in favor of the bridge?<BR>\n" );
document.write( "E) Are gender and opinion about the bridge mutually exclusive events? Why?<BR>\n" );
document.write( "F) Are gender and opinion about the bridge independent events? Why? Show some \"proof\"<BR>\n" );
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document.write( "Solution:<BR>\n" );
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document.write( "The question gives a contingency table, from which we are expected to extract answers to the given questions.<BR>\n" );
document.write( "We will define events:<BR>\n" );
document.write( "O=opposition to project<BR>\n" );
document.write( "~O=in favour of project<BR>\n" );
document.write( "M=opinion by a male<BR>\n" );
document.write( "~M=opinion by a female<BR>\n" );
document.write( "We assume all samples are randomly selected, and size of sample is large (so that there is no repetition of selected subjects).<BR>\n" );
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document.write( "(a) <BR>\n" );
document.write( "P(O)=total number of oppositions / total number of subjects asked.<BR>\n" );
document.write( "= 3920/5600<BR>\n" );
document.write( "= 7/10<BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "P(O&cap;M)<BR>\n" );
document.write( "=840/5600<BR>\n" );
document.write( "= 3/20<BR>\n" );
document.write( "(c)<BR>\n" );
document.write( "P(O&cup;M)<BR>\n" );
document.write( "=P(O)+P(M)-P(M&cap;O)<BR>\n" );
document.write( "=(3920+2240-840)/5600<BR>\n" );
document.write( "= 19/20<BR>\n" );
document.write( "(d)<BR>\n" );
document.write( "P(~O|M)<BR>\n" );
document.write( "=P(~O&cap;M)/P(M)<BR>\n" );
document.write( "=(1400/5600)/(2240/5600)<BR>\n" );
document.write( "=1400/2240<BR>\n" );
document.write( "= 5/8<BR>\n" );
document.write( "(e)<BR>\n" );
document.write( "They are not mutually exclusive events, since every opinion is associated with a gender.<BR>\n" );
document.write( "(f)<BR>\n" );
document.write( "Gender and opinion are not independent events.<BR>\n" );
document.write( "If they were independent, then P(M&cap;O)=P(M)*P(O)<BR>\n" );
document.write( "Here P(M&cap;O)=840/5600=3/20<BR>\n" );
document.write( "P(M)*P(O)=2240/5600*3920/5600=7/25<BR>\n" );
document.write( "Since they are not equal, the two characteristics are not independent.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );