Algebra.Com's Answer #842298 by ikleyn(52814)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Twenty percent of the employees of a large company are female. Use the normal approximation
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document.write( "of the binomial probabilities to answer the following questions. What is the probability
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document.write( "that in a random sample of 80 employees
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document.write( "(a) exactly 16 will be female?
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document.write( "(b) 14 or more will be female?
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document.write( "(c) 15 or fewer will be female?
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document.write( "(d) 18 or more will be female
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document.write( "(e) exactly 17 will be female?
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document.write( "You have a binomial distribution with large number of trials n = 80 \r\n" );
document.write( "and individual probability of success p = 20% = 0.2.\r\n" );
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document.write( "You want to approximate it by the normal distribution.\r\n" );
document.write( "You should use the mean value m = p*n = 0.2*80 = 16 and standard deviation \r\n" );
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document.write( " SD =  =  = 3.57771.\r\n" );
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document.write( "Also, you should use the continuity correction factor.\r\n" );
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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" );
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document.write( "https://www.statology.org/normal-approximation\r\n" );
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document.write( "https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/\r\n" );
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document.write( "https://online.stat.psu.edu/stat414/lesson/28/28.1\r\n" );
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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" );
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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" );
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document.write( "(a) In this case, you should find the area under the normal curve between z-scores 15.5 and 16.5\r\n" );
document.write( " (using the correction factor)\r\n" );
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document.write( " z1 z2 mean SD <<<---=== formatting pattern\r\n" );
document.write( " P(x = 16) = normcdf(15.5, 16.5, 16, 3.57771) = 0.1111 (rounded).\r\n" );
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document.write( "(b) In this case, you should find the area under the normal curve on the right from 13.5\r\n" );
document.write( " (using the correction factor)\r\n" );
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document.write( " z1 z2 mean SD <<<---=== formatting pattern\r\n" );
document.write( " P(x >= 14) = normcdf(13.5, 9999, 16 3.57771) = 0.7577 (rounded). \r\n" );
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document.write( "(c) In this case, you should calculate the area under the normal curve on the left of the mark z = 15.5\r\n" );
document.write( " (using the correction factor)\r\n" );
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document.write( " z1 z2 mean SD <<<---=== formatting pattern\r\n" );
document.write( " P(x <= 15) = normcdf(-9999, 15.5, 16, 3.57771) = 0.4444 (rounded). \r\n" );
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document.write( "(d) Do it as (b).\r\n" );
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document.write( "(e) Do it as (a).\r\n" );
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document.write( "Solved.\r
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document.write( "If you are a beginner student in learning probability distributions, I advise you to start learning
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document.write( "this kind of computations using the online calculator, to which I referred above.\r
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document.write( "It provides a graphical support, so at each step you do understand what you are doing.
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document.write( "In addition, this graphical support prevents you from making mistakes.\r
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document.write( "When you learn enough this kind of computations, you can switch to your regular calculator,
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document.write( "but even then you may use the online calculator for checking purposes.\r
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