document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1128376>Question 1128376</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Several students are unprepared for multiple choice quiz with 10 questions and all their answers are guesses each question has five possible answers and only one of them is correct <BR>\n" );
document.write( "A) find the mean and standard deviation for the number of correct answers for such students <BR>\n" );
document.write( "B) would it be unusual such a student to pass by guessing and getting at least 7 correct answers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #744839 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=744839\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Part A)\r<BR>\n" );
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document.write( "n = 10<BR>\n" );
document.write( "p = probability of getting a question right<BR>\n" );
document.write( "p = 1/5 = 0.2\r<BR>\n" );
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document.write( "mean = n*p<BR>\n" );
document.write( "mean = 10*0.2<BR>\n" );
document.write( "mean = 2\r<BR>\n" );
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document.write( "standard deviation = sqrt(n*p*(1-p))<BR>\n" );
document.write( "standard deviation = sqrt(10*0.2*(1-0.2))<BR>\n" );
document.write( "standard deviation = sqrt(10*0.2*0.8)<BR>\n" );
document.write( "standard deviation = sqrt(1.6)<BR>\n" );
document.write( "standard deviation = 1.2649<BR>\n" );
document.write( "This is approximate to four decimal places\r<BR>\n" );
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document.write( "=============================================================================\r<BR>\n" );
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document.write( "Part B)\r<BR>\n" );
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document.write( "Compute the binomial probability of getting exactly 7 questions correct<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "10 C 7 = (10!)/(7!*(10-7)!)<BR>\n" );
document.write( "10 C 7 = 120<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 7) = (10 C 7)*(0.2)^(7)*(1-0.2)^(10-7)<BR>\n" );
document.write( "P(X = 7) = (120)*(0.2)^(7)*(0.8)^(3)<BR>\n" );
document.write( "P(X = 7) = 0.000786432\r<BR>\n" );
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document.write( "Compute the binomial probability of getting exactly 8 questions correct<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "10 C 8 = (10!)/(8!*(10-8)!)<BR>\n" );
document.write( "10 C 8 = 45<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 8) = (10 C 8)*(0.2)^(8)*(1-0.2)^(10-8)<BR>\n" );
document.write( "P(X = 8) = (45)*(0.2)^(8)*(0.8)^(2)<BR>\n" );
document.write( "P(X = 8) = 0.000073728\r<BR>\n" );
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document.write( "Compute the binomial probability of getting exactly 9 questions correct<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "10 C 9 = (10!)/(9!*(10-9)!)<BR>\n" );
document.write( "10 C 9 = 10<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 9) = (10 C 9)*(0.2)^(9)*(1-0.2)^(10-9)<BR>\n" );
document.write( "P(X = 9) = (10)*(0.2)^(9)*(0.8)^(1)<BR>\n" );
document.write( "P(X = 9) = 0.000004096\r<BR>\n" );
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document.write( "Compute the binomial probability of getting exactly 10 questions correct<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "10 C 10 = (10!)/(10!*(10-10)!)<BR>\n" );
document.write( "10 C 10 = 1<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 10) = (10 C 10)*(0.2)^(10)*(1-0.2)^(10-10)<BR>\n" );
document.write( "P(X = 10) = (1)*(0.2)^(10)*(0.8)^(0)<BR>\n" );
document.write( "P(X = 10) = 0.0000001024\r<BR>\n" );
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document.write( "Add up the four probabilities found above to find the probability of getting 7 or more correct. <BR>\n" );
document.write( "P(X >= 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)<BR>\n" );
document.write( "P(X >= 7) = 0.000786432+0.000073728+0.000004096+0.0000001024 <BR>\n" );
document.write( "P(X >= 7) = 0.0008643584\r<BR>\n" );
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document.write( "The probability of getting 7 or more correct answers is 0.0008643584<BR>\n" );
document.write( "This is a very small value so it's a very unusual (aka unlikely) event, which is what we expect for any student who randomly guesses at the answers. <BR>\n" );
document.write( "So it would be very unusual for a student to pass if they were to randomly guess at all the questions.</font> </SPAN>\n" );
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