Algebra.Com's Answer #831322 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "750 eggs are randomly sampled from a population where 14% of all eggs are fertilized.
\n" );
document.write( "Use the normal approximation to the binomial to find the following probabilities rounded to 3 decimal places.\r
\n" );
document.write( "\n" );
document.write( "a. Find the probability that exactly 106 of the eggs are fertilized. Correct
\n" );
document.write( "b. Find the probability that at least 106 of the eggs are fertilized. Incorrect
\n" );
document.write( "c. Find the probability that fewer than 106 of the eggs are fertilized. Incorrect
\n" );
document.write( "d. Find the probability that between 104 and 106, inclusive, of the eggs are fertilized.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "You have a binomial distribution with large number of trials n = 750 and \r\n" );
document.write( "individual probability of success p = 0.14.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You want to approximate it by the normal distribution.\r\n" );
document.write( "You should use the mean value m = p*n = 0.14*750 = 105 and standard deviation \r\n" );
document.write( "\r\n" );
document.write( " SD =  =  = 9.5026.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Also, you should use the continuity correction factor.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "About approximation of the binomial distribution by normal distribution and continuity correction factor \r\n" );
document.write( "see your textbook and/or these Internet sources\r\n" );
document.write( "\r\n" );
document.write( "https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/\r\n" );
document.write( "\r\n" );
document.write( "https://online.stat.psu.edu/stat414/lesson/28/28.1\r\n" );
document.write( "\r\n" );
document.write( "https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " For calculations, you may use your calculator (function normcdf), or Excel spreadsheet (function NORMDIST);\r\n" );
document.write( " or online calculator https://onlinestatbook.com/2/calculators/normal_dist.html\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(a) In this case, you should calculate the area under the normal curve between 105.5 and 106.5 \r\n" );
document.write( " (using the correction factor)\r\n" );
document.write( "\r\n" );
document.write( " P(x = 106) = normcdf(105.5, 106.5, 105, 9.5026)= 0.0417 (rounded)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b) In this case, you should calculate the area under the normal curve on the right from 105.5\r\n" );
document.write( " (using the correction factor)\r\n" );
document.write( "\r\n" );
document.write( " P(x >= 106) = normcdf(105.5, 9999, 105, 9.5026) = 0.479 (rounded).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c) In this case, you should calculate the area under the normal curve on the left from 105.5\r\n" );
document.write( " (using the correction factor)\r\n" );
document.write( "\r\n" );
document.write( " P(x < 106) = normcdf(-9999, 105.5, 105, 9.5026) = 0.521 (rounded). \r\n" );
document.write( "\r\n" );
document.write( " Notice that it is the complement to the value of probability in (b)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(d) In this case, you should calculate the area under the normal curve between 103.5 and 106.5\r\n" );
document.write( " (using the correction factor)\r\n" );
document.write( "\r\n" );
document.write( " P(104 <= x <= 106) = normcdf(103.5, 106.5, 105, 9.5026) = 0.125 (rounded). \r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "---------------\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "If you are a beginner student in learning probability distribution, I advise you to start learning
\n" );
document.write( "this kind of computations using the online calculator, to which I referred above.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "It provides a graphical support, so at each step you do understand what you are doing.
\n" );
document.write( "In addition, this graphical support prevents you from making mistakes.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "When you learm enough this kind of computations, you can switch to your regular calculator,
\n" );
document.write( "but even then you may use the online calculator for checking purposes.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |