document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192102>Question 1192102</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch a normal curve for each distribution. Label the x-axis at one, two, and three standard deviations from the mean.\r<BR>\n" );
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document.write( "1. Mean = 30; Standard deviation = 5\r<BR>\n" );
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document.write( "2. Mean = 95; Standard deviation = 12 </SPAN>\n" );
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document.write( "I'll do problem 1 to get you started.\r<BR>\n" );
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document.write( "Draw a bell curve as shown below. At the very center, where the hill is at its peak, we have the mean = 30.\r<BR>\n" );
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document.write( "The gap between adjacent tickmarks is a gap of 1 standard deviation, which in this case is 5 units. <BR>\n" );
document.write( "In other words, the gap between any adjacent tickmarks is 5 units.\r<BR>\n" );
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document.write( "If we started at the center (mean = 30) and moved 1 tickmark or standard deviation to the right, then we arrive at 30+5 = 35.<BR>\n" );
document.write( "Another tickmark over and we arrive at 30+2*5 = 40<BR>\n" );
document.write( "and so on\r<BR>\n" );
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document.write( "The same idea applies in reverse when going to the left.<BR>\n" );
document.write( "Start at mean = 30 and move 1 tickmark to the left to get to 30-5 = 25<BR>\n" );
document.write( "Then another tick over and we get to 30-2*5 = 20<BR>\n" );
document.write( "and so on.\r<BR>\n" );
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document.write( "Standard convention is to do 3 standard deviations away from the mean (which accounts for roughly 99.7% of the normally distributed population; according to the Empirical Rule)<BR>\n" );
document.write( "This accounts for the following tickmarks on the x axis:<BR>\n" );
document.write( "15, 20, 25, 30, 35, 40, 45\r<BR>\n" );
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document.write( "This is what the final sketch should look like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/ihSBc8x.png\"><BR>\n" );
document.write( "I used GeoGebra to make the figure, which is free software. I encourage this route as well, or any free online software that offers similar capabilities. <BR>\n" );
document.write( "Though if your teacher wants you to sketch by hand, then be sure to follow those instructions of course.\r<BR>\n" );
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document.write( "If you wanted, you can add in the labels shown in blue<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/p9is99V.png\"><BR>\n" );
document.write( "to get a better sense of what each location is.<BR>\n" );
document.write( "Something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu%2B2%2Asigma ALIGN=MIDDLE  ALT=\"mu%2B2%2Asigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > means we're 2 standard deviations above the mean<BR>\n" );
document.write( "While another example like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+-+3%2Asigma ALIGN=MIDDLE  ALT=\"mu+-+3%2Asigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > indicates we're now 3 standard deviations below the mean. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" > = greek letter mu = mean<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > = greek letter sigma = standard deviation\r<BR>\n" );
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document.write( "Optionally you can add in vertical lines to help better separate the various pieces or sections.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/b2bhsR7.png\"><BR>\n" );
document.write( "The vertical lines also help show the markers of each standard deviation distance from the center mean.\r<BR>\n" );
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document.write( "Side note: the vertical lines at 15 and 45 are barely noticeable. These locations are 3 standard deviations away from the mean.<BR>\n" );
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