document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1142656>Question 1142656</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, this is a statistics problem. Is there a formula that I would use to find the minimum and maximum values below?  \r<BR>\n" );
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document.write( "Use the following information to determine your answers: The typical amount of sleep per night that adults get has a bell-shaped distribution with a mean of 7.5 hours and a standard deviation of 1.3 hours.\r<BR>\n" );
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document.write( "About 68% of adults typically sleep between a minimum of ___ hours a night and a maximum of ____ hours a night. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #763440 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=763440\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the typical amount of sleep per night has a mean of 7.5 hours and a standard deviation of 1.3 hours.\r<BR>\n" );
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document.write( "about 68% of adults typically sleep between a minimum of x hours a night and a maximum of y hours a night.\r<BR>\n" );
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document.write( "you want to find the value of x and y.\r<BR>\n" );
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document.write( "you need to find the z-score of the low end of the 68% and you need to find the z-score of the high end of the 68%.\r<BR>\n" );
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document.write( "the 68% is assumed to be in the middle of the normal distribution curve.<BR>\n" );
document.write( "there would be two tails:<BR>\n" );
document.write( "one on the left of the low end z-score.<BR>\n" );
document.write( "the other on the right of the high end z-score.\r<BR>\n" );
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document.write( "you can use the z-score tables, or you can use a z-score calculator.<BR>\n" );
document.write( "the use of a z-score calculator is the easiest and the most accurate.<BR>\n" );
document.write( "usually, accuracy to 2 decimal places is sufficient.<BR>\n" );
document.write( "the z-score tables only give you z-score rounded to 2 decimal places.<BR>\n" );
document.write( "if you need more accuracy, then you would need to interpolate.<BR>\n" );
document.write( "the calculators give you a minimum of rounding to 3 decimal places, i believe.\r<BR>\n" );
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document.write( "one such calculator can be found at <a  rel=nofollow HREF= \"\" target= \"_blank\"></a>.\r<BR>\n" );
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document.write( "your area under the normal distribution curve is .68.<BR>\n" );
document.write( "your tails on each end would be (1 -.68) / 2 = .32/2 = .16.<BR>\n" );
document.write( "the tail on the lower end is to the left of the low end z-score.<BR>\n" );
document.write( "the tail on the upper end is to the right of the high end z-score.\r<BR>\n" );
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document.write( "since the normal distribution curve is symmetric about the mean, if you find the low end z-score, then the high end z-score will be the same value except with a different sign.\r<BR>\n" );
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document.write( "so, you use the online calculator to find the low end z-score.\r<BR>\n" );
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document.write( "you are looking for a z-score that has p(Z < x) equal to .16.<BR>\n" );
document.write( "that means that the area to the left of the low end z-score is .16.<BR>\n" );
document.write( "the calculator tells you that the low end z-score is minus .994 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "since the normal distribution is symmetric about the mean, your high end z-score should be plus .994.\r<BR>\n" );
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document.write( "you can test to see if this is true by looking for the area of .16 to the right of the high end z-score.\r<BR>\n" );
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document.write( "since this calculator only gives you the area to the left of the z-score, you would take .16 and subtract it from 1 to get .84.\r<BR>\n" );
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document.write( "a z-score with an area of .84 to the left of it is the same z-score with an area of .16 to the right of it.\r<BR>\n" );
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document.write( "use the calculator again to get z = plus .994\r<BR>\n" );
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document.write( "this confirms the symmetry of the normal distribution curve about the mean.\r<BR>\n" );
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document.write( "you have a low end z-score of minus .994.<BR>\n" );
document.write( "you have a high end z-score of plus .995.\r<BR>\n" );
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document.write( "the next step is to calculate the raw score associated with the z-score.\r<BR>\n" );
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document.write( "the z-score formula is z = (x-m)/s\r<BR>\n" );
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document.write( "z is the z-score.<BR>\n" );
document.write( "x is the raw score.<BR>\n" );
document.write( "m is the mean.<BR>\n" );
document.write( "s is the standard deviation, in this case.\r<BR>\n" );
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document.write( "in this problem, m is equal to 7.5 and s is equal to 1.3.\r<BR>\n" );
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document.write( "the low end z-score formula becomes -.994 = (x-7.5)/1.3.<BR>\n" );
document.write( "solve for x to get x = -.994 * 1.3 + 7.5 = 6.2078\r<BR>\n" );
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document.write( "the high end z-score formula becomes .994 = (2-7.5)/1.3<BR>\n" );
document.write( "solve for x to get x = .994 * 1.3 + 7.5 = 8.7922.\r<BR>\n" );
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document.write( "visually, this looks like this, using another online calculator that is more sophisticated and also gives you a graphical presentation.\r<BR>\n" );
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document.write( "that calculator can be found at <a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "the following displays were used.\r<BR>\n" );
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document.write( "the first two are from the stattrek calculator.<BR>\n" );
document.write( "the third display is from the david m. lane calculator using z-scores.<BR>\n" );
document.write( "the fourth display is from the david m .alen calculator suing raw scores.\r<BR>\n" );
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