document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202686>Question 1202686</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 56.7 inches, and standard deviation of 6.7 inches.\r<BR>\n" );
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document.write( "What is the probability that the height of a randomly chosen child is between 41.65 and 50.85 inches? Do not round until you get your your final answer, and then round to 3 decimal places.<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 #837703 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=837703\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer:   <font color=red size=4>0.180</font> (approximate)<BR>\n" );
document.write( "Due to rounding error, a slightly different value may be the answer your teacher wants (eg: 0.179).\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "mu = 56.7 = population mean<BR>\n" );
document.write( "sigma = 6.7 = population standard deviation\r<BR>\n" );
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document.write( "Compute the z score when x = 41.65 <BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (41.65 - 56.7)/6.7<BR>\n" );
document.write( "z = -2.246269<BR>\n" );
document.write( "z = -2.25\r<BR>\n" );
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document.write( "Do the same for x = 50.85<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (50.85 - 56.7)/6.7<BR>\n" );
document.write( "z = -0.873134<BR>\n" );
document.write( "z = -0.87\r<BR>\n" );
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document.write( "The task of computing P(41.65 < x < 50.85) is roughly equivalent to computing P(-2.25 < z < -0.87)\r<BR>\n" );
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document.write( "Now I'll use a Z table. <BR>\n" );
document.write( "This kind of table should be found in the appendix section of your textbook.\r<BR>\n" );
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document.write( "If you don't have your textbook with you, then you can refer to a table such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "That page provides examples of how to read that table. Let me know if you have questions about it.\r<BR>\n" );
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document.write( "Use such a table to find that,<BR>\n" );
document.write( "P(Z < -2.25) = 0.01222<BR>\n" );
document.write( "and<BR>\n" );
document.write( "P(Z < -0.87) = 0.19215\r<BR>\n" );
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document.write( "Subtract the two results to compute the area between those z scores.<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-2.25 < Z < -0.87) = P(Z < -0.87) - P(Z < -2.25)<BR>\n" );
document.write( "P(-2.25 < Z < -0.87) = 0.19215 - 0.01222<BR>\n" );
document.write( "P(-2.25 < Z < -0.87) = 0.17993\r<BR>\n" );
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document.write( "That leads us to P(41.65 < x < 50.85) = 0.17993\r<BR>\n" );
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document.write( "Roughly 17.993% of the children have a height between 41.65 inches and 50.85 inches\r<BR>\n" );
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document.write( "0.17993 then rounds to <font color=red>0.180</font> when rounding to three decimal places.\r<BR>\n" );
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document.write( "Depending on which table you use, or if you used a stats calculator, your final answer may be slightly different due to rounding error. \r<BR>\n" );
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document.write( "This link shown here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "offers a way to compute the answer fairly quickly. It also provides a nice diagram.<BR>\n" );
document.write( "When I used that calculator, I got approximately 0.1789 which rounds to 0.179. That's not too far from the 0.180 I got earlier. \r<BR>\n" );
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document.write( "Here's an article talking about the TI84 and the normal cdf function<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/normal-probabilities-ti-84-calculator/\">https://www.statology.org/normal-probabilities-ti-84-calculator/</a><BR>\n" );
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