document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168066>Question 1168066</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particular fruit's weights are normally distributed, with a mean of 756 grams and a standard deviation of 14 grams.\r<BR>\n" );
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document.write( "The heaviest 20% of fruits weigh more than how many grams?\r<BR>\n" );
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document.write( "Give your answer to the nearest gram. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792676 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=792676\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the mean is 756 grams.<BR>\n" );
document.write( "the standard deviation is 14 grams\r<BR>\n" );
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document.write( "look into the z-score table.\r<BR>\n" );
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document.write( "you want to find the z-score with an area of .2 to the right of it.<BR>\n" );
document.write( "the z-score table only shows the area to the left of the z-score.<BR>\n" );
document.write( "subtract .2 from 1 to get .8.<BR>\n" );
document.write( "you want to find the z-core with an area of .8 to the left of it.<BR>\n" );
document.write( "that will be the same z-score with an area of .2 to the right of it.\r<BR>\n" );
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document.write( "the z-score table that i use can be found at <a  rel=nofollow HREF= \"https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf\" target = \"_blank\">https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf</a>\r<BR>\n" );
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document.write( "looking into the table, i found:\r<BR>\n" );
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document.write( "an area of .79955 is associated with a z-score of .84<BR>\n" );
document.write( "an area of .80234 is associated with a z-score of .85\r<BR>\n" );
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document.write( "the difference between those areas is .00279 and the difference between those z-scores is .01\r<BR>\n" );
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document.write( "take the highest area and subtract .2 from it to get a difference of .00234.<BR>\n" );
document.write( "divide that difference from the overall difference of .00279 to get a ratio of .00234 / .00279 = .8387096774.<BR>\n" );
document.write( "multiply the difference between the two z-scores of .01 to get .01 * .8387096774 = .0083870968.<BR>\n" );
document.write( "subtract .008370968 from the z-score of .85 to get a z-score of .85 - .008370968 = .841612903323.\r<BR>\n" );
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document.write( "that's a reasonably close approximation of what the real z-score should be.<BR>\n" );
document.write( "you can also use a z-score calculator that will also get you presumably more accurate answer with a lot less headaches.\r<BR>\n" );
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document.write( "the z-score calculator on the TI-84 Plus that i have gave me a z-score of .84162122335.<BR>\n" );
document.write( "the z-scores are the same out to 4 decimal digits which is usually good enough.\r<BR>\n" );
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document.write( "anyway, now that you have the z-score, you can solve for the raw score.\r<BR>\n" );
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document.write( "the z-score formula is:\r<BR>\n" );
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document.write( "z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation or the standard error\r<BR>\n" );
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document.write( "in this case, s is the standard deviation.\r<BR>\n" );
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document.write( "replace m and s and z with their respective values to solve for x.\r<BR>\n" );
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document.write( "you get .841612903323 = (x - 756) / 14.<BR>\n" );
document.write( "solve for x to get x = .841612903323 * 14 + 756 = 767.7825806.\r<BR>\n" );
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document.write( "80% of the raw scores will be less than that and 20% of the raw scores will be higher than that.\r<BR>\n" );
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document.write( "that can be seen through the use of the following calculator that can be found at <a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/normal-distribution\" target = \"_blank\">https://www.omnicalculator.com/statistics/normal-distribution</a>\r<BR>\n" );
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document.write( "here's a display of the results of using that calculator.\r<BR>\n" );
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document.write( "my inputs were mean of 756 standard deviation of 14 and p(x > X) of .2\r<BR>\n" );
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document.write( "the calculator filled in the rest.\r<BR>\n" );
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document.write( "it told me that the value of X was 767.783.\r<BR>\n" );
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document.write( "that's very close to the raw score i calculated manually of 767.783 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "unless you have to use the tables and do the manual interpolation, the recommendation is to use the calculator.\r<BR>\n" );
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document.write( "it's much easier.\r<BR>\n" );
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document.write( "anyway, round the raw score to the nearest gram to get 768 grams as your answer.\r<BR>\n" );
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document.write( "fyi, i used the z-score that i manually calculated through interpolation.<BR>\n" );
document.write( "in most cases, i would just use the z-score i got through the use of a calculator.<BR>\n" );
document.write( "i only did it manually to show you how the interpolation works.<BR>\n" );
document.write( "if you use the tables, you could also just use the z-score associated with the area closest to the one that you want and that will be good enough.<BR>\n" );
document.write( "your instructor should be your guide as to how close an answer you need to get.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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