document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204642>Question 1204642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> U.S. internet users spend an average of<BR>\n" );
document.write( "18.3 hours a week online. If 95% of users spend between 13.1 and 23.5 hours a week, what is the probability that a randomly selected user is online less than 15 hours a week? </SPAN>\n" );
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document.write( "To solve this problem, we need to use the properties of the normal distribution.\r<BR>\n" );
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document.write( "First, we need to calculate the standard deviation (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of the distribution. We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95 ALIGN=MIDDLE  ALT=\"95\"   DISABLED_style= \"max-width:90%; height:auto;\" >% of the data lies between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.1 ALIGN=MIDDLE  ALT=\"13.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23.5 ALIGN=MIDDLE  ALT=\"23.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours. <BR>\n" );
document.write( "This range corresponds to the interval from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.96 ALIGN=MIDDLE  ALT=\"-1.96\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.96 ALIGN=MIDDLE  ALT=\"1.96\"   DISABLED_style= \"max-width:90%; height:auto;\" > standard deviations from the mean in a normal distribution (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95 ALIGN=MIDDLE  ALT=\"95\"   DISABLED_style= \"max-width:90%; height:auto;\" >% of the data in a normal distribution is within <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.96+ ALIGN=MIDDLE  ALT=\"1.96+\"   DISABLED_style= \"max-width:90%; height:auto;\" >standard deviations of the mean).\r<BR>\n" );
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document.write( "We can set up the following equations to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+-+1.96sigma=+13.1 ALIGN=MIDDLE  ALT=\"mu+-+1.96sigma=+13.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+%2B+1.96sigma+=+23.5 ALIGN=MIDDLE  ALT=\"mu+%2B+1.96sigma+=+23.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mu=13.1%2B23.5 ALIGN=MIDDLE  ALT=\"2mu=13.1%2B23.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mu=36.6 ALIGN=MIDDLE  ALT=\"2mu=36.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu=18.3 ALIGN=MIDDLE  ALT=\"mu=18.3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma=%2823.5-18.3%29%2F1.96 ALIGN=MIDDLE  ALT=\"sigma=%2823.5-18.3%29%2F1.96\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma=2.65 ALIGN=MIDDLE  ALT=\"sigma=2.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Next, we need to calculate the z-score for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X=15 ALIGN=MIDDLE  ALT=\"X=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=%28X-mu%29%2Fsigma ALIGN=MIDDLE  ALT=\"z=%28X-mu%29%2Fsigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=%2815-18.3%29%2F2.65 ALIGN=MIDDLE  ALT=\"z=%2815-18.3%29%2F2.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-1.245 ALIGN=MIDDLE  ALT=\"z=-1.245\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The z-score tells us how many standard deviations an element is from the mean. A z-score of <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.245 ALIGN=MIDDLE  ALT=\"-1.245\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that 15 hours is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.245 ALIGN=MIDDLE  ALT=\"1.245\"   DISABLED_style= \"max-width:90%; height:auto;\" > standard deviations below the mean.\r<BR>\n" );
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document.write( "Finally, we need to find the probability that a randomly selected user is online less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15+ ALIGN=MIDDLE  ALT=\"15+\"   DISABLED_style= \"max-width:90%; height:auto;\" >hours a week. This is the same as finding the area to the left of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1.245 ALIGN=MIDDLE  ALT=\"z+=+-1.245\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the standard normal distribution.\r<BR>\n" );
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document.write( "You can find this value in a standard normal distribution table or use a calculator with a normal distribution function. \r<BR>\n" );
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document.write( "probabilities for the normal distribution:<BR>\n" );
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document.write( "endpoint: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "mean: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18.3 ALIGN=MIDDLE  ALT=\"18.3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "standard deviation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.65 ALIGN=MIDDLE  ALT=\"2.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3C15%29+=+0.1065 ALIGN=MIDDLE  ALT=\"P%28X%3C15%29+=+0.1065\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, the probability that a randomly selected user is online less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15+ ALIGN=MIDDLE  ALT=\"15+\"   DISABLED_style= \"max-width:90%; height:auto;\" >hours a week is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.1065 ALIGN=MIDDLE  ALT=\"0.1065\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.65 ALIGN=MIDDLE  ALT=\"10.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >%.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );