document.write( "Question 519049: 3. The weights of men are normally distributed with a mean of 200 pounds and a standard deviation of 25 pounds. \r
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Algebra.Com's Answer #345372 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Although there are many normal curves, they all share an important property\r\n" );
document.write( "that allows us to treat them in a uniform fashion. \r\n" );
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document.write( "The 68-95-99.7% Rule\r\n" );
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document.write( "All normal density curves satisfy the following property which is often\r\n" );
document.write( "referred to as the Empirical Rule. \r\n" );
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document.write( "approximately 68% \r\n" );
document.write( "of the observations fall within 1 standard deviation of the mean, that is,\r\n" );
document.write( "between \"mu-sigma%29+and+%7B%7B%7Bmu%2Bsigma\". \r\n" );
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document.write( "approximately 95% \r\n" );
document.write( "of the observations fall within 2 standard deviations of the mean, that is,\r\n" );
document.write( "between \"mu-2sigma%29+and+%7B%7B%7Bmu%2B2sigma\". \r\n" );
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document.write( "approximately 99.7% \r\n" );
document.write( "of the observations fall within 3 standard deviations of the mean, that is,\r\n" );
document.write( "between \"mu-3sigma%29+and+%7B%7B%7Bmu%2B3sigma\".\r\n" );
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document.write( "Thus, for a normal distribution, almost all values lie within 3 standard\r\n" );
document.write( "deviations of the mean.\r\n" );
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document.write( "Below are the percentages of observations that fall between \r\n" );
document.write( "the intervals:\r\n" );
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document.write( "3. The weights of men are normally distributed with a mean of 200 pounds\r\n" );
document.write( "and a standard deviation of 25 pounds. \r\n" );
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document.write( "So let's calculate the values on the x-axis: \r\n" );
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document.write( "\"mu-3sigma=200-3%2A25=200-75=125\"\r\n" );
document.write( "\"mu-2sigma=200-2%2A25=200-50=150\"\r\n" );
document.write( "\"mu-sigma=200-25=175\"\r\n" );
document.write( "\"mu-200\"\r\n" );
document.write( "\"mu%2Bsigma=200%2B25=225\"\r\n" );
document.write( "\"mu%2B2sigma=200%2B50=250\"\r\n" );
document.write( "\"mu%2B3sigma=200%2B75=275\"\r\n" );
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document.write( "and re-draw the figure with these values:\r\n" );
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document.write( "a. Find the percent of men with weights between 175 pounds and 225 pounds.\r\n" );
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document.write( "Add the percentages in the graph above in the two regions\r\n" );
document.write( "between 175 and 225:  34.134% + 34.134% = 68.268%.  Or use the 68-95-97.5 \r\n" );
document.write( "rule and get about 68% because 175 = \"mu-sigma\" and 225 = \"mu%2Bsigma\".\r\n" );
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document.write( "b. Among 100 men how many are expected to weigh \r\n" );
document.write( "1) Between 175 pounds and 225 pounds?\r\n" );
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document.write( "Since that's about 68%, we find that 68% of 100 men is 68 men.\r\n" );
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document.write( "b. Among 100 men how many are expected to weigh\r\n" );
document.write( "2) Less than or equal 200 pounds?\r\n" );
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document.write( "One way is to add the percentages in the graph above in all the regions\r\n" );
document.write( "to the left of the mean: 0.135%+2.140%+13.591%+34.134% = 50%.\r\n" );
document.write( "Another way is to just observe that half or 50% are below the mean.\r\n" );
document.write( "Either way, 50% of 100 men is 50 men\r\n" );
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document.write( "b. Among 100 men how many are expected to weigh\r\n" );
document.write( "3) Greater than equal 250 pounds?\r\n" );
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document.write( "I think you might have meant this one to be 250 instead of 240, so I\r\n" );
document.write( "chnaged it to 250.  To do it for 240 pounds involves z-scores and a\r\n" );
document.write( "normal table, and since the other ones just required the 68-95-99.7% \r\n" );
document.write( "rule, I'll assume you meant this to be 250, not 240.  so we add the\r\n" );
document.write( "percentages in the two regions the the right of 250 which are \r\n" );
document.write( "2.140%+0.135% = 2.275% and 2.275% of 100 men is 2.275 men, but sinch\r\n" );
document.write( "we can't cut men into fractions, we round to about 2 men.\r\n" );
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document.write( "Now if you did mean 240, then the z-score is 1.6, and so you look up\r\n" );
document.write( "z=1.6 in the z-table you either find 0.4452 or 0.9452, depending on\r\n" );
document.write( "which type of tabel you are using.\r\n" );
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document.write( "If you found 0.4452, you subtract 0.5-0.4452 = 0.0548\r\n" );
document.write( "If you found 0.9452, you subtract 1.0-0.4452 = 0.0548\r\n" );
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document.write( "So if it really is 240 and not 250, then the answer is about\r\n" );
document.write( "5.48% of 100, or 5.48 which rounds to about 5 men.\r\n" );
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document.write( "Edwin
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