document.write( "Question 306452: - Use Chebyshev’s theorem to find what percent of the values will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14. \r
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Algebra.Com's Answer #219372 by Edwin McCravy(20060)\"\" \"About 
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document.write( "Chebyshev's theorem:  The fraction of any set of numbers \r\n" );
document.write( "lying within k standard deviations of those numbers of \r\n" );
document.write( "the mean of those numbers is at least\r\n" );
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document.write( "\"1-1%2Fk%5E2\"\r\n" );
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document.write( "where \"k=%22the_%27within%27_number%22%2F%22the_standard_deviation%22\", \r\n" );
document.write( "which must be greater than 1.\r\n" );
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\n" ); document.write( "Use Chebyshev’s theorem to find what percent of the values
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document.write( "We subtract 151-123 and get 28, which tells us that\r\n" );
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document.write( "123 is 28 units below the mean\r\n" );
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document.write( "We subtract 179-151 and also get 28, which tells us that\r\n" );
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document.write( "151 is 28 units above the mean.\r\n" );
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document.write( "Those two together tell us that the values between 123 and 179 are \r\n" );
document.write( "all within 28 units of the mean.  Therefore the \"within number\"\r\n" );
document.write( "is 28.\r\n" );
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document.write( "So we find the number of standard deviations, k, which the \r\n" );
document.write( "\"within number\", 28, amounts to by dividing it by the standard deviation:\r\n" );
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document.write( "\"k=%22the_%27within%27_number%22%2F%22the_standard_deviation%22+=+28%2F14=2\"\r\n" );
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document.write( "So now we know that the values between 123 and 179 are \r\n" );
document.write( "all within 28 units of the mean, which is the same as within \r\n" );
document.write( "k=2 standard deviations of the mean.\r\n" );
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document.write( "Now, since k > 1 we can use Chebyshev's formula to find the fraction of \r\n" );
document.write( "the data that are within k=2 standard deviations of the mean.  Substituting\r\n" );
document.write( "k=2 we have:\r\n" );
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document.write( " \"1-1%2Fk%5E2=1-1%2F2%5E2+=+1-1%2F4=3%2F4\"\r\n" );
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document.write( "So \"3%2F4\" of the data lie between 123 and 179.  And since\r\n" );
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document.write( "\"3%2F4+=+%2275%25%22\" that implies that 75% of the data values are\r\n" );
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document.write( "between 123 and 179.  \r\n" );
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document.write( "Edwin
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