document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204356>Question 1204356</A>: <I><SPAN STYLE=\"word-break: break-all;\"> As reported in \"Runner's World\" magazine, the times of the finishers in the New York City 10 km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let x denote finishing time for the finishers.<BR>\n" );
document.write( "a) The distribution of the variable x has mean <BR>\n" );
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document.write( "b) The distribution of the standardized variable z has mean <BR>\n" );
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document.write( "c) The percentage of finishers with times between 40 and 65 minutes is equal to the area under the standard normal curve between <BR>\n" );
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document.write( "d) The percentage of finishers with times exceeding 86 minutes is equal to the area under the standard normal curve that lies to the <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 #840725 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=840725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean is 61.<BR>\n" );
document.write( "standard deviation is 9.\r<BR>\n" );
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document.write( "excerpt from the web:<BR>\n" );
document.write( "A standardized variable (sometimes called a z-score or a standard score) is a variable that has been rescaled to have a mean of zero and a standard deviation of one.\r<BR>\n" );
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document.write( "formula for z-score is z = (x-m)/s<BR>\n" );
document.write( "x is the variable.<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "z is the standardized variable, called the z-score.<BR>\n" );
document.write( "the standardization allows different sets of data with different means and standard deviations to be compared to each other.<BR>\n" );
document.write( "the z-score tells you how many standard deviations you are from the mean.\r<BR>\n" );
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document.write( "here's a reference.<BR>\n" );
document.write( "click on all the pages to oget the full story.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://statistics.laerd.com/statistical-guides/standard-score.php\" target = \"_blank\">https://statistics.laerd.com/statistical-guides/standard-score.php</a>\r<BR>\n" );
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document.write( "the mean of the variable is 61 and the standard deviation is 9.\r<BR>\n" );
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document.write( "the mean of the standardized variable is 0 and the standard deviation is 1.\r<BR>\n" );
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document.write( "z-score formula is z = (x-m)/s<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the variable score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "when x is 40, z = (40 - 61) / 9 = -2.3333.<BR>\n" );
document.write( "when x is 65, z = (65 - 61) / 9 = .44444.\r<BR>\n" );
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document.write( "the percentage of finishers with times between 40 and 65 minutes is equal to the area under the normal distribution curve between z-scores of -2.3333 and .4444.<BR>\n" );
document.write( "the z-scores are usualy rounded to between 2 and 4 decimal digits.<BR>\n" );
document.write( "there are calculators to help you find the area in bgetween directly.<BR>\n" );
document.write( "otherwisee you would find the area to the left of the low z-score and to the left of the high z-score and subtract the smaller area from the larger area to get the area in between.\r<BR>\n" );
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document.write( "to find the percentages of finishers that exceed 86 minutes, find the z-score for x = 86 and then find the area to the right of that.<BR>\n" );
document.write( "z = (86 - 61) / 9 = 2.7778.<BR>\n" );
document.write( "find the area to the right of that.\r<BR>\n" );
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document.write( "i rounded all z-score to 4 decimal digits which provides sufficient accuracy for most situations.\r<BR>\n" );
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document.write( "a calculator that's very useful can be found at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "this calculator allows you to find values from the raw scores and from the z-scores.<BR>\n" );
document.write( "i'll show you the results from it that were obtained from the z-scores amd from the raw scpres.<BR>\n" );
document.write( "they'll either be the same or close to each other, depending on rounding differences.\r<BR>\n" );
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document.write( "first is finding the area in between 40 and 65 using z-score and then raw scores.\r<BR>\n" );
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document.write( "next is finding the area to the right of 86 using z-scores and then raw scores.\r<BR>\n" );
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document.write( "in the calculator, when you're working with z-score, you set the mean to 0 and the standard deviation to 1; when you're working with raw scores, you set the mean and standard deviatiion to whatever they are for the problem you are working.\r<BR>\n" );
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document.write( "answers to your questions.\r<BR>\n" );
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document.write( "a) The distribution of the variable x has mean of 61 and standard deviation of 9.\r<BR>\n" );
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document.write( "b) The distribution of the standardized variable z has meanof 0 and standard deviation of 1.\r<BR>\n" );
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document.write( "c) The percentage of finishers with times between 40 and 65 minutes is equal to the area under the standard normal curve between z = -2.3333 and .4444.\r<BR>\n" );
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document.write( "d) The percentage of finishers with times exceeding 86 minutes is equal to the area under the standard normal curve that lies to the right of z-score of 2.7778. </SPAN>\n" );
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