document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205471>Question 1205471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Twenty percent of the employees of a large company are female. Use the normal approximation <BR>\n" );
document.write( "of the binomial probabilities to answer the following questions. What is the probability <BR>\n" );
document.write( "that in a random sample of 80 employees<BR>\n" );
document.write( "a. exactly 16 will be female? <BR>\n" );
document.write( "b. 14 or more will be female?<BR>\n" );
document.write( "c. 15 or fewer will be female? <BR>\n" );
document.write( "d. 18 or more will be female<BR>\n" );
document.write( "e. exactly 17 will be female? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842297 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842297\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Check out this article<BR>\n" );
document.write( "https://www.statology.org/normal-approximation/\r<BR>\n" );
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document.write( "If you're still stuck then move to the next section shown below.\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "n = sample size = 80<BR>\n" );
document.write( "p = probability of selecting a woman = 0.20\r<BR>\n" );
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document.write( "Computing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=np ALIGN=MIDDLE  ALT=\"np\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%281-p%29 ALIGN=MIDDLE  ALT=\"n%281-p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to 16 and 64 respectively.<BR>\n" );
document.write( "Both results are larger than 5, so they meet the criteria <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=np+%3E=+5 ALIGN=MIDDLE  ALT=\"np+%3E=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%281-p%29+%3E=+5 ALIGN=MIDDLE  ALT=\"n%281-p%29+%3E=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Some stats textbooks will use 10 in place of 5; luckily 16 and 64 meet that threshold as well.<BR>\n" );
document.write( "We confirm that a normal approximation to the binomial can be used.<BR>\n" );
document.write( "Make sure to do this check before going any further. \r<BR>\n" );
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document.write( "mu = mean<BR>\n" );
document.write( "sigma = standard deviation\r<BR>\n" );
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document.write( "mu = n*p<BR>\n" );
document.write( "mu = 80*0.20<BR>\n" );
document.write( "mu = 16\r<BR>\n" );
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document.write( "sigma = sqrt(n*p*(1-p))<BR>\n" );
document.write( "sigma = sqrt(80*0.20*(1-0.20))<BR>\n" );
document.write( "sigma = 3.577708764 approximately\r<BR>\n" );
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document.write( "Refer to the link I posted to account for the continuity correction. This is where we go from a discrete setting to a continuous setting.\r<BR>\n" );
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document.write( "x = 16 leads to the interval 15.5 < x < 16.5\r<BR>\n" );
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document.write( "Use any stats calculator you want to find the area under the normal distribution curve between those endpoints.<BR>\n" );
document.write( "I'll use a TI84.\r<BR>\n" );
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document.write( "If you are using that as well then press the button labeled \"2nd\".<BR>\n" );
document.write( "Then press the VARS key.<BR>\n" );
document.write( "Move to <font color=green>normalCDF</font>\r<BR>\n" );
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document.write( "The full command to type in would be <font color=green>normalCDF(15.5, 16.5, 16, 3.577708764)</font><BR>\n" );
document.write( "The standard template is <font color=green>normalCDF(L, R, mu, sigma)</font><BR>\n" );
document.write( "L = left endpoint<BR>\n" );
document.write( "R = right endpoint\r<BR>\n" );
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document.write( "The result of the calculation <font color=green>normalCDF(15.5, 16.5, 16, 3.577708764)</font> is <font color=red>approximately 0.11114589</font><BR>\n" );
document.write( "Round that value however needed.\r<BR>\n" );
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document.write( "Therefore, <BR>\n" );
document.write( "P(15.5 < x < 16.5) = <font color=red>0.11114589</font> approximately.\r<BR>\n" );
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document.write( "There's about an 11.1146% chance of having 16 women in the sample of 80 employees.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "\"14 or more\" translates to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+14 ALIGN=MIDDLE  ALT=\"x+%3E=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Revisit the continuity correction chart in this link<BR>\n" );
document.write( "https://www.statology.org/normal-approximation/\r<BR>\n" );
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document.write( "Apply the continuity correction to make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+14 ALIGN=MIDDLE  ALT=\"x+%3E=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" > convert to x > 13.5<BR>\n" );
document.write( "We subtract off 0.5 from 14 to get this new endpoint.\r<BR>\n" );
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document.write( "Using the TI84 we could type in<BR>\n" );
document.write( "<font color=green>normalCDF(13.5, 9999, 16, 3.577708764)</font><BR>\n" );
document.write( "Notice there technically isn't a right endpoint. <BR>\n" );
document.write( "The 9999 is some really large number to effectively stand in for positive infinity.\r<BR>\n" );
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document.write( "The TI84 calculator says the result is roughly <font color=red>0.75765257</font>\r<BR>\n" );
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document.write( "Another calculator you could use is this<BR>\n" );
document.write( "https://davidmlane.com/normal.html<BR>\n" );
document.write( "It offers a diagram to go along with the area value.\r<BR>\n" );
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document.write( "Another alternative is the <a  rel=nofollow HREF=\"https://wiki.geogebra.org/en/Normal_Command\">Normal</a> command in GeoGebra. <BR>\n" );
document.write( "Or you could use the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/normdist-function-126db625-c53e-4591-9a22-c9ff422d6d58\">normDist</a> command in a spreadsheet.<BR>\n" );
document.write( "There are many other calculators that provide the same functionality.\r<BR>\n" );
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document.write( "If your teacher wants you to use a Z table instead of a calculator, then be sure to follow those instructions. You'll need to convert to a z score first before using the table. <BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Use the continuity correction chart to go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+15 ALIGN=MIDDLE  ALT=\"x+%3C=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > to x < 15.5\r<BR>\n" );
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document.write( "Whichever calculator or table you end up using, the answer should be approximately <font color=red>0.44442708</font><BR>\n" );
document.write( "The answer may slightly vary depending on what method you used.\r<BR>\n" );
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document.write( "I'll leave the other parts for the student to do.<BR>\n" );
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