document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=248978>Question 248978</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here is a simple probability model for multiple-choice tests. Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A good student has a higher p than a poor student.) The correctness of an answer to any specific question doesn't depend on other questions. A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion  from an SRS of size n drawn from a population with population proportion p. \r<BR>\n" );
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document.write( "(a) Julie is a good student for whome p = 0.73. Find the probability that Julie scores 67% or lower on a 110 question test.\r<BR>\n" );
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document.write( "(c) How many questions must the test contain in order to reduce the standard deviation of Julie's proportion of correct answers to one-fourth its value for an 100-item test?<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 #181393 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=181393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is a simple probability model for multiple-choice tests. <BR>\n" );
document.write( "Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possible questions. <BR>\n" );
document.write( "(A good student has a higher p than a poor student.) <BR>\n" );
document.write( "The correctness of an answer to any specific question doesn't depend on other questions. <BR>\n" );
document.write( "A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion from an SRS of size n drawn from a population with population proportion p.<BR>\n" );
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document.write( "(a) Julie is a good student for whom p = 0.73. Find the probability that Julie scores 67% or lower on a 110 question test.<BR>\n" );
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document.write( "mean = 0.73<BR>\n" );
document.write( "std = sqrt{0.73*0.27/110] = 0.0391<BR>\n" );
document.write( "z(0.67) = (0.67-0.73)/0.0391 = -1.5345\r<BR>\n" );
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document.write( "P(0 <- phat <= 0.67) = P(-inf < z < -1.5345) = 0.0625<BR>\n" );
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document.write( "(c) How many questions must the test contain in order to reduce the standard deviation of Julie's proportion of correct answers to one-fourth its value for an 100-item test?<BR>\n" );
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document.write( "Since std = sqrt(pq/n)<BR>\n" );
document.write( "(1/4)sqrt(pq/n) = sqrt(pq/16n)<BR>\n" );
document.write( "16*110 = 1760<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.\r<BR>\n" );
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