document.write( "Question 1185601: When appointing computer operators a firm requires that the candidates pass a
\n" ); document.write( "written examination. The examination paper contains 100 multiple choice questions, each with a
\n" ); document.write( "choice of three answers of which only one is correct. A pass is obtained by answering 40 or more
\n" ); document.write( "questions correctly. Use the Normal distribution as an approximation to the binomial distribution
\n" ); document.write( "to estimate the probability that a candidate who chooses the answer to each question randomly
\n" ); document.write( "will pass the examinations. \n" ); document.write( "

Algebra.Com's Answer #849567 by ikleyn(52943) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "When appointing computer operators a firm requires that the candidates pass a
\n" ); document.write( "written examination. The examination paper contains 100 multiple choice questions, each with a
\n" ); document.write( "choice of three answers of which only one is correct. A pass is obtained by answering 40 or more
\n" ); document.write( "questions correctly. Use the Normal distribution as an approximation to the binomial distribution
\n" ); document.write( "to estimate the probability that a candidate who chooses the answer to each question randomly
\n" ); document.write( "will pass the examinations.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The original problem is on Binomial distribution with n = 100 trials and  p = 1/3 for individual \r\n" );
document.write( "success  (p= 1/3 to answer correctly and q = 1-p = 2/3 to answer incorrectly).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, formally there are standard formulas for binomial distribution, but the Math does not \r\n" );
document.write( "recommend to use it, because the formulas include high degree exponents and too great binomial\r\n" );
document.write( "coefficients.  Instead, there is another, very effective and standard approach on solving such problems.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This approach is to use an appropriate normal distribution as an approximation to the Binomial distribution.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The relevant normal distribution has the mean  m = n*p =  = 33.33333\r\n" );
document.write( "and the standard deviation  S =  =  = 4.714.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This normal curve covers all 100 questions on the horizontal axis, and we should take the area under\r\n" );
document.write( "this normal curve on the right of the threshold value 40.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "A convenient tool to find the area of interest is to use the standard function  normcdf \r\n" );
document.write( "(which means Cumulative Distrifution Functionfor normal distribution).  \r\n" );
document.write( "This function is in any regular calculator like TI-83/84.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, you write/print in your calculator\r\n" );
document.write( "\r\n" );
document.write( "                 z1     z2      mean     SD     <<<---===  formatting pattern.\r\n" );
document.write( "    P = normcdf(39.5,  9999,  33.33333, 4.714)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Here z1 = 39.5  stands for 40 and presents the continuing correction;  z2 stand for infinity on the horizontal axis,\r\n" );
document.write( "mean is the mean and SD is the standard deviation.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the function will return the probability, which you are looking for: P = 0.0954.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is the ANSWER to the problem's question  P = 0.0954, obtained with the normal distribution approximation.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved, with explanations.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you want to learn more from the Internet about the normal distribution as an approximation for binomial distribution,\r
\n" ); document.write( "\n" ); document.write( "go to Google with these keywords \"normal distribution as an approximation for binomial distribution\" and learn from there.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );