document.write( "Question 319777: \"Scores on a test are normally distributed with a mean of 61 and a standard deviation of 10.2 Find which separates the bottom 81% from the top 19%\" \n" ); document.write( "
Algebra.Com's Answer #229031 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "If you have a TI-83 or 84,\r\n" );
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document.write( "From the cleared main screen\r\n" );
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document.write( "Press 2ND\r\n" );
document.write( "Press VARS\r\n" );
document.write( "Press 3\r\n" );
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document.write( "you will see this on the screen -->  invNorm(\r\n" );
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document.write( "after that type .81,61,10.2) so that you see this on the\r\n" );
document.write( "screen --->  invNorm(.81,61,10.2)\r\n" );
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document.write( "[If you can't find the comma key it is the key just to the right of x2).\r\n" );
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document.write( "Press ENTER\r\n" );
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document.write( "You will see 69.95454216, which you round to the nearest whole \r\n" );
document.write( "number 70.\r\n" );
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document.write( "If you are not allowed to use a calculator, then you must go through\r\n" );
document.write( "the following long process:\r\n" );
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document.write( "Draw a normal curve on the x (not z) axis marking the middle point\r\n" );
document.write( "as the mean 61.  Then mark off units right and left by adding the\r\n" );
document.write( "standard deviation 10.2 to the mean and the standard deviation to that\r\n" );
document.write( "to mark points to the right and subtracting 10.2 from the mean to get\r\n" );
document.write( "points on the left side. \r\n" );
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document.write( "Next we redraw the normal curve with a z-axis of z-scores instead \r\n" );
document.write( "of an x-axis of \"x-scores\" (actual values).  This graph has z=0 \r\n" );
document.write( "corresponding to x=61, z=1 corresponding to x=71.2, z=2 corresponding \r\n" );
document.write( "to x=81.4, z=-1 corresponding to 50.8, and z=-2 corresponding to 40.6.  \r\n" );
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document.write( "So you see that the z-score is just the number of standard deviations\r\n" );
document.write( "the corresponding x-score is from the mean.\r\n" );
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document.write( "The area under the entire normal curve is 1.  So the area under the\r\n" );
document.write( "left side of the normal curve is .5. We want to find the value of z\r\n" );
document.write( "such that 81% or .81 of the area is under the curve to the left of that\r\n" );
document.write( "z-value.  So .5 of the .81 is to the left of z=0 and that leaves\r\n" );
document.write( ".81 - .5 = .31 for the area on the right of z=0. So we look in the body\r\n" );
document.write( "of the table and find the nearest value to .31 for the area to the\r\n" );
document.write( "right of z=0 that we need to make up the rest of the 81%.  We find \r\n" );
document.write( "that .3106 is the closest area value in the table.  Then we observe that\r\n" );
document.write( ".3106 occurs where z = .88. So that is the cut off value we are looking\r\n" );
document.write( "for.  I'll indicate it with a green line at z=.88, just a little to the\r\n" );
document.write( "left of z=1.\r\n" );
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document.write( "The area under the curve to the left of the green line represents the bottom\r\n" );
document.write( "81% and the area under the curve to the right of the green line represents\r\n" );
document.write( "the top 19%.\r\n" );
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document.write( "Now what we have to do is to find the x-score on the x-axis that corresponds\r\n" );
document.write( "to the z-score of .88 on the z-axis.  I'll redraw the normal curve with the\r\n" );
document.write( "x-axis, keeping the green dividing line:\r\n" );
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document.write( "We can see that the green dividing line is a little to the left of x=71.2.\r\n" );
document.write( "We calculate that value by using the formula\r\n" );
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document.write( "\"z\"\"%22=%22\"\"%28x-mu%29%2Fsigma\"\r\n" );
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document.write( "which when solved for x is\r\n" );
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document.write( "\"x\"\"%22=%22\"\"mu%2Bsigma%2Az\"\r\n" );
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document.write( "So the x-score that corresponds to the z-score .88 is\r\n" );
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document.write( "\"x\"\"%22=%22\"\"61%2B10.2%2A.88\"\r\n" );
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document.write( "\"x\"\"%22=%22\"\"69.976\"\r\n" );
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document.write( "[Notice that is slightly different from what the calculator gave.\r\n" );
document.write( "But that's because we could not find .31 exactly in the table so\r\n" );
document.write( "the best we could do was to use .3106 instead.  The calculator \r\n" );
document.write( "is more accurate than the table.\r\n" );
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document.write( "However, rounding that to the nearest whole number gives us the\r\n" );
document.write( "same cut-off score of 70.\r\n" );
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document.write( "Edwin

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