document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019132>Question 1019132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A class is taking a multiple-choice quiz that consists of 15 questions.  Each question has 3 answer options: A, B, and C. The instructor made a mistake and printed the quiz for a much more advanced class. The students don’t know any of the content and they randomly guess at every question.\r<BR>\n" );
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document.write( "Explain why this is a binomial random.\r<BR>\n" );
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document.write( "What are the values of the parameters n and p?\r<BR>\n" );
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document.write( "What is the mean quiz score?\r<BR>\n" );
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document.write( "What is the standard deviation of the quiz scores?\r<BR>\n" );
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document.write( "In order to pass, a student needs to get 9 or more questions correct.  What is the probability that a student will get 9 or more questions correct?\r<BR>\n" );
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document.write( "What is the probability that a student will get exactly 9 questions correct?\r<BR>\n" );
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document.write( "What is the probability that a student will get all 15 questions correct?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635132 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=635132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "A class is taking a multiple-choice quiz that consists of 15 questions.  Each question has 3 answer options: A, B, and C. The instructor made a mistake and printed the quiz for a much more advanced class. The students don’t know any of the content and they randomly guess at every question.<BR>\n" );
document.write( "a. Explain why this is a binomial random.<BR>\n" );
document.write( "b. What are the values of the parameters n and p?<BR>\n" );
document.write( "c. What is the mean quiz score?<BR>\n" );
document.write( "d. What is the standard deviation of the quiz scores?<BR>\n" );
document.write( "e. In order to pass, a student needs to get 9 or more questions correct.  <BR>\n" );
document.write( "   What is the probability that a student will get 9 or more questions correct?<BR>\n" );
document.write( "f. What is the probability that a student will get exactly 9 questions correct?<BR>\n" );
document.write( "g. What is the probability that a student will get all 15 questions correct?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "a.<BR>\n" );
document.write( "A situation must satisfy the following conditions in order that the binomial distribution be applicable.<BR>\n" );
document.write( "1. The experiment must consist of a fixed and known number of trials, each of which must be random and independent of each other.<BR>\n" );
document.write( "2. Each trial must be a Bernoulli trial, i.e. it must have exactly two outcomes, true or false, or success or failure.<BR>\n" );
document.write( "3. The probability of success must be constant throughout the trials.<BR>\n" );
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document.write( "Since the given situation satisfies all of the requirements,<BR>\n" );
document.write( "1. all answers are random guesses, and naturally, they are independent of each other.  There are exactly 15 trials in the experiment.<BR>\n" );
document.write( "2. answers to multiple choice questions must be either right or wrong, hence Bernoulli trial.<BR>\n" );
document.write( "3. The probability of success is known (1/3) and is constant throughout the experiment (15 questions=15 trials).  <BR>\n" );
document.write( "The Binomial Distribution may be used to model the situation.<BR>\n" );
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document.write( "b.<BR>\n" );
document.write( "parameters <BR>\n" );
document.write( "n=number of trials = 15<BR>\n" );
document.write( "p=probability of success in each trial = 1/3<BR>\n" );
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document.write( "c.<BR>\n" );
document.write( "Mean, &mu;, of a binomial distribution = np = 15*(1/3) = 5<BR>\n" );
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document.write( "d. <BR>\n" );
document.write( "variance, &sigma;&sup2; = np(1-p) = 15*(1/3)(1-1/3) = 10/3<BR>\n" );
document.write( "standard deviation, &sigma; = &radic;(10/3) = 1.826 approx.<BR>\n" );
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document.write( "e.<BR>\n" );
document.write( "the probability of having X questions correct is given by the probability mass function (PMF)<BR>\n" );
document.write( "P(X;n;p)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2CX%29%2Ap%5Ex%2A%281-p%29%5E%281-x%29 ALIGN=MIDDLE  ALT=\"C%28n%2CX%29%2Ap%5Ex%2A%281-p%29%5E%281-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2815%2CX%29%2A%281%2F3%29%5EX%2A%281-1%2F3%29%5E%2815-X%29 ALIGN=MIDDLE  ALT=\"C%2815%2CX%29%2A%281%2F3%29%5EX%2A%281-1%2F3%29%5E%2815-X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To get 9 or more questions correct, we can sum <BR>\n" );
document.write( "&Sigma;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+C%2815%2Ci%29%2A%281%2F3%29%5Ei%2A%282%2F3%29%5E%2815-i%29 ALIGN=MIDDLE  ALT=\"+C%2815%2Ci%29%2A%281%2F3%29%5Ei%2A%282%2F3%29%5E%2815-i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for i=9 to 15.<BR>\n" );
document.write( "= 0.02232+0.00670+0.00025+0.00003+0.00000+0.00000<BR>\n" );
document.write( "=0.03083<BR>\n" );
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document.write( "f.<BR>\n" );
document.write( "P(9;15;1/3)=0.02232   (from part (e))<BR>\n" );
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document.write( "g.<BR>\n" );
document.write( "P(15;15,1/3)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.9692%2A10%5E%28-8%29 ALIGN=MIDDLE  ALT=\"6.9692%2A10%5E%28-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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