document.write( "Question 1167969: The owner of a fish market determined that the mean weight for a catfish is 3.2 pounds assuming that the weight of a catfish follows a normal distribution and its standard deviation is unknown. He also knew that that probability of a randomly selected catfish that would weigh more than 3.8 pounds is 20% and the probability that a randomly selected catfish that would weigh less than 2.8 pounds is 30%. What is the probability that a randomly selected catfish will weigh between 2.6 and 3.6 pounds?\r
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Algebra.Com's Answer #851733 by ikleyn(52908) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The owner of a fish market determined that the mean weight for a catfish is 3.2 pounds.
\n" ); document.write( "Assuming that the weight of a catfish follows a normal distribution and its standard deviation is unknown.
\n" ); document.write( "He also knew that that probability of a randomly selected catfish that would weigh more than 3.8 pounds is 20%
\n" ); document.write( "and the probability that a randomly selected catfish that would weigh less than 2.8 pounds is 30%.
\n" ); document.write( "What is the probability that a randomly selected catfish will weigh between 2.6 and 3.6 pounds?\r
\n" ); document.write( "\n" ); document.write( "I know that the answer is 50% or 0.5 but I am having trouble figuring out how to get to that answer.
\n" ); document.write( "I think it is normal distribution but I don't know where to begin.
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\n" ); document.write( "\n" ); document.write( "        I am very glad to find this problem. Usually, in this circle of problems, all problems are \r
\n" ); document.write( "\n" ); document.write( "        typical and the whole clearing is trampled, so it is a rare event to find something unordinary.\r
\n" ); document.write( "\n" ); document.write( "        While this one is a truly unordinary (although is not complicated).\r
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document.write( "Let's draw the coordinate axis and mark there the given numbers.\r\n" );
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document.write( "The coordinate axis represents the weight of catfish.  \r\n" );
document.write( "The mark 3.2 is the mean (the average weight), in pounds.\r\n" );
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document.write( "We are given in the problem that              20% of the fish weigh is on the right of the mark 3.8 pounds,\r\n" );
document.write( "and                                           30% of the fish weight is on the left  of the mark 2.8 pounds.\r\n" );
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document.write( "       Notice that in this problem, the fish weight is always the area under the normal curve \r\n" );
document.write( "    between the given marks, or on the left of the given mark,  or on the right of the given mark.\r\n" );
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document.write( "Using the mean, it is the same as to say that 20% of the fish weight is on the right of the mark 3.2+0.6 pounds,\r\n" );
document.write( "and                                           30% of the fish weight is on the left  of the mark 3.2-0.4 pounds.\r\n" );
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document.write( "So, 50% of the fish weight is in the interval    (3.2-0.4,3.2+0.6)  pounds.\r\n" );
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document.write( "Now, the question relates to the interval (2.6,    3.6) pounds.\r\n" );
document.write( "This interval is the same as              (3.2-0.6,3.2+0.4) pounds.\r\n" );
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document.write( "Thus, we know that 50% of the fish weight is in the interval           (-0.4,+0.6)  around the mean,\r\n" );
document.write( "and the question is what percent of the fish weight is in the interval (-0.6,+0.4)  around the mean.\r\n" );
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document.write( "Due to the symmetry of the normal distribution curve, the answer is that\r\n" );
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document.write( "    the percent of the fish weight in the interval (-0.6,+0.4) around the mean \r\n" );
document.write( "    is the same 50% as in the interval             (-0.4,+0.6) around the mean.\r\n" );
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document.write( "It is the ANSWER to the problem's question.\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, the problem is solved in full.\r
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\n" ); document.write( "\n" ); document.write( "The key to the solution is that the interval under the question
\n" ); document.write( "is SYMMETRIC to the interval GIVEN in the problem.\r
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\n" ); document.write( "\n" ); document.write( "You posted an ELEGANT problem to the forum and get an ELEGANT solution back.\r
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\n" ); document.write( "\n" ); document.write( "My congratulations ( ! )\r
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