document.write( "Question 599968: Chebychev's Theorem:
\n" ); document.write( "Suppose the time needed to assemble a piece of furniture is not normally distributed and that the mean assembly time is 28 minutes. What is the value of the standard deviation if at lease 77% of the assembly times are between 24 and 32 minutes? \n" ); document.write( "
Algebra.Com's Answer #379140 by Edwin McCravy(20067)\"\" \"About 
You can put this solution on YOUR website!
Suppose the time needed to assemble a piece of furniture is not normally distributed and that the mean assembly time is 28 minutes. What is the value of the standard deviation if at lease 77% of the assembly times are between 24 and 32 minutes?
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document.write( "The mean is 28 minutes.  Therefore\r\n" );
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document.write( "24 minutes is 4 minutes below the mean and 32 minutes is 4 minutes\r\n" );
document.write( "above the mean,  So we are talking about 4 minutes from the mean.\r\n" );
document.write( "We want to know what standard deviation would guarantee that at least\r\n" );
document.write( "77% of the data would lie within 4 minutes of the mean.\r\n" );
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document.write( "Chebychev's theorem says that 100(1-\"1%2Fk%5E2\")% of the data lies within\r\n" );
document.write( "k standard deviations of the mean.  That means that ks = 4, where s is \r\n" );
document.write( "the standard deviation.\r\n" );
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document.write( "So first find k by setting 100(1-\"1%2Fk%5E2\")% equal to 77% and solve \r\n" );
document.write( "for k:\r\n" );
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document.write( "100(1-\"1%2Fk%5E2\")% = 77%\r\n" );
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document.write( "Drop the percents:\r\n" );
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document.write( "100(1-\"1%2Fk%5E2\") = 77\r\n" );
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document.write( "Divide both sides by 100\r\n" );
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document.write( "1 - \"1%2Fk%5E2\" = .77    <--(If your book just gives 1-\"1%2Fk%5E2\" you can start here)\r\n" );
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document.write( "Clear of fractions by multiplying through by kČ:\r\n" );
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document.write( "         kČ - 1 = .77kČ\r\n" );
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document.write( "Get the kČ terms on the left and the 1 on the right:\r\n" );
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document.write( "     kČ - .77kČ = 1\r\n" );
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document.write( "Factor out kČ on the left:\r\n" );
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document.write( "    kČ(1 - .77) = 1\r\n" );
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document.write( "        kČ(.23) = 1\r\n" );
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document.write( "Divide both sides by .23\r\n" );
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document.write( "             kČ = \"1%2F.23\"\r\n" );
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document.write( "             kČ = 4.347826087\r\n" );
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document.write( "Take the square root of both sides:\r\n" );
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document.write( "              k = 2.085144141\r\n" );
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document.write( "So that means that at least 77% of the data lies within\r\n" );
document.write( "2.085144141 standard deviations of the mean. \r\n" );
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document.write( "       Since ks = 4,\r\n" );
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document.write( "   2.085144141s = 4 minutes\r\n" );
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document.write( "Divide both sides by 2.085144141\r\n" );
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document.write( "              s = \"4%2F2.085144141\" minutes\r\n" );
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document.write( "              s = 1.918332609 minutes\r\n" );
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document.write( "Round off to however many decimal places your teacher told you,\r\n" );
document.write( "probably to 1.9 or 1.92 minutes.\r\n" );
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document.write( "Edwin
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