document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202163>Question 1202163</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The incubation time for chicks is normally distributed with mean 21 days and standard deviation approximately 1 day. If 1000 eggs are being incubated, how many chicks do we expect will hatch?\r<BR>\n" );
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document.write( "a) in 19 to 23 days? \r<BR>\n" );
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document.write( "b) in 22 days or more? \r<BR>\n" );
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document.write( "c) in 21 days or fewer? \r<BR>\n" );
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document.write( "d) in 18 to 21 days? \r<BR>\n" );
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document.write( "e) in 19 days or fewer \r<BR>\n" );
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document.write( "f) in 19 to 24 days? \r<BR>\n" );
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document.write( "g) in 18 to 24 days? \r<BR>\n" );
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document.write( "h) in 18 days or more? \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836807 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=836807\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll do parts (a), (b), and (c) to get you started.\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "mu = 21 = population mean<BR>\n" );
document.write( "sigma = 1 = population standard deviation<BR>\n" );
document.write( "x = incubation time in days\r<BR>\n" );
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document.write( "Find the z score when x = 19<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (19 - 21)/1<BR>\n" );
document.write( "z = -2\r<BR>\n" );
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document.write( "Repeat for x = 23 to get z = 2\r<BR>\n" );
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document.write( "The task of finding P(19 < x < 23) is the same as P(-2 < z < 2).\r<BR>\n" );
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document.write( "Now use a table such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "Similar tables are found in the back of your stats textbook.\r<BR>\n" );
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document.write( "An alternative is to use a stats calculator such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "However, I'll stick to using the table.\r<BR>\n" );
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document.write( "That table says:<BR>\n" );
document.write( "P(Z < -2) = 0.02275<BR>\n" );
document.write( "and<BR>\n" );
document.write( "P(Z < 2) = 0.97725<BR>\n" );
document.write( "Those values are approximate.\r<BR>\n" );
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document.write( "This gives<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-2 < Z < 2) = P(Z < 2) - P(Z < -2)<BR>\n" );
document.write( "P(-2 < Z < 2) = 0.97725 - 0.02275<BR>\n" );
document.write( "P(-2 < Z < 2) = 0.9545<BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "About 95.45% of the population is between the z scores z = -2 and z = 2.<BR>\n" );
document.write( "This translates to roughly 95.45% of the population of eggs with incubation times between x = 19 and x = 23 days.\r<BR>\n" );
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document.write( "If there are 1000 eggs being incubated, then about 0.9545*1000 = 954.5 = 955 eggs will hatch in the timeframe of 19 days to 23 days.\r<BR>\n" );
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document.write( "Note that the Empirical Rule says roughly 95% of the population is within 2 standard deviations.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>955</font>\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( "Find the z score when x = 22<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (22 - 21)/1<BR>\n" );
document.write( "z = 1<BR>\n" );
document.write( "We are 1 standard deviation above the mean.\r<BR>\n" );
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document.write( "Use the z table to find that<BR>\n" );
document.write( "P(Z < 1.00) = 0.84134<BR>\n" );
document.write( "which must mean<BR>\n" );
document.write( "P(Z > 1.00) = 1-P(Z < 1.00)<BR>\n" );
document.write( "P(Z > 1.00) = 1-0.84134<BR>\n" );
document.write( "P(Z > 1.00) = 0.15866\r<BR>\n" );
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document.write( "About 15.866% of the population has an incubation time of 22 days or more.\r<BR>\n" );
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document.write( "15.866% of 1000 = 0.15866*1000 = 158.66 = 159\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>159</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "We could convert to a z score and use a table, but that might be a bit overkill here. <BR>\n" );
document.write( "50% of the population is always below the mean of a normal distribution.\r<BR>\n" );
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document.write( "50% of 1000 = 0.50*1000 = 500\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>500</font><BR>\n" );
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