document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1054112>Question 1054112</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here is a simple probability model for multiple-choice tests. Suppose each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) Answers to different questions are independent. (Round your answers to four decimal places.)<BR>\n" );
document.write( "(a) Stacey is a good student for whom p = 0.71. Use the Normal approximation to find the probability that Stacey scores between (and including) 70% and 80% on a 100-question test. <BR>\n" );
document.write( "(b) If the test contains 250 questions, what is the probability that Stacey will score between (and including) 70% and 80%? You see that Stacey's score on the longer test is more likely to be close to her \"true score.\" <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 #669314 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=669314\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a) Stacey is a good student for whom p = 0.71. Use the Normal approximation to find the probability that Stacey scores between (and including) 70% and 80% on a 100-question test.<BR>\n" );
document.write( "mean = n*p = 100*0.71 = 71<BR>\n" );
document.write( "std = sqrt(n*p*q) = sqrt(71*0.29) = 4.5376<BR>\n" );
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document.write( "Binomial: P(70<= x <=80) ~ Normal:P(69.5<= x <=80.5)<BR>\n" );
document.write( "z(69.5) = (59.5-71)/4.5376 = -2.5344<BR>\n" );
document.write( "z(80.5) = (80.5-71)/4.5376 = 2.0936<BR>\n" );
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document.write( "Ans: p(-2.7344 < z < 2.0936) = normalcdf(-2.7344,2.0936) = 0.9762<BR>\n" );
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document.write( "(b) If the test contains 250 questions, what is the probability that Stacey will score between (and including) 70% and 80%? You see that Stacey's score on the longer test is more likely to be close to her \"true score.\" <BR>\n" );
document.write( "mean = n*p = 250*0.71 = 177.5<BR>\n" );
document.write( "std = sqrt(177.5*0.29) = 7.1746<BR>\n" );
document.write( "Note: 0.7*250 = 175 ; 0.8*250 = 200<BR>\n" );
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document.write( "z(175) = (175-177.5)/7.1746 = -2.5<BR>\n" );
document.write( "z(200) = (200-177.5)/7.1746 = 3.1363<BR>\n" );
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document.write( "etc.<BR>\n" );
document.write( "I'll let you finish the problem.<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
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