document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122524>Question 1122524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The mean weight of a box of cereal filled by a machine is 16.0 ounces, with a standard deviation of 0.5 ounce. If the weights of all the boxes filled by the machine are normally distributed, what percent of the boxes will weigh the following amounts?\r<BR>\n" );
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document.write( "a. less than 15.5 ounces\r<BR>\n" );
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document.write( "b.between 15.8 and 16.2 ounces\r<BR>\n" );
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document.write( "I would like to see the steps if possible </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #738776 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=738776\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean = 16 ounces.<BR>\n" );
document.write( "standard deviation = .5 ounces.<BR>\n" );
document.write( "weights are normally distributed.<BR>\n" );
document.write( "what percent will weight:<BR>\n" );
document.write( "a.  less than 15.5 ounces.<BR>\n" );
document.write( "b.  between 15.8 and 16.2 ounces.\r<BR>\n" );
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document.write( "there are calculators that make this easy, but i will show you how to do it manually using just the normal distribution table.\r<BR>\n" );
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document.write( "you need to find the z-scores before using the z-score tables.\r<BR>\n" );
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document.write( "the formula for the z-score 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 you are comparing against the mean.<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "in your problem, ....\r<BR>\n" );
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document.write( "m = 26<BR>\n" );
document.write( "s = .5\r<BR>\n" );
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document.write( "for part a, your raw score is 15.5.\r<BR>\n" );
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document.write( "formula becomes z = (15.5 - 16) / .5<BR>\n" );
document.write( "result is z-score = -1.\r<BR>\n" );
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document.write( "look up the z-score of -1.0 in the z-score table to find that the area to the left of that z-score is .1587.\r<BR>\n" );
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document.write( "that means that 15.87% of the boxes will weight less than 15.5 ounces.\r<BR>\n" );
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document.write( "for part b, you will be looking for the z-scores associated with a raw score of 15.8 and a raw score of 16.2.\r<BR>\n" );
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document.write( "formula for 15.8 becomes z = (15.8 - 16) / .5.<BR>\n" );
document.write( "result is z-score = -.4\r<BR>\n" );
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document.write( "formula for 16.2 becomes z = (16.2 - 16) / .5.<BR>\n" );
document.write( "result is z-score = .4\r<BR>\n" );
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document.write( "you will look up the area to the left of a z-score of -.4 and to the left of a z-score of .4.\r<BR>\n" );
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document.write( "those areas will be .3446 to the left of -.4 and .6554 to the left of .4\r<BR>\n" );
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document.write( "subtract the smaller area from the larger area to get .3108 which tells you that 31.08% of the scores will be between 15.8 and 16.2 ounces.\r<BR>\n" );
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document.write( "there is an online calculator that i use to check my results.\r<BR>\n" );
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document.write( "that calculator can be found at <a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "using z-scores, that calculator confirms the results from the table.\r<BR>\n" );
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document.write( "the z-score results from that calculator are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090701.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090702.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "using raw scores, that calculator confirms the results from the z-scores.\r<BR>\n" );
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document.write( "the raw score results from that calculator are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090703.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090704.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "results from the table i used are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090706.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "that table can be found at <a  rel=nofollow HREF= \"http://www.z-table.com/\" target = \"_blank\">http://www.z-table.com/</a>\r<BR>\n" );
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document.write( "when you are using the table, the first column is the integer part the the tenth part of the z-score.\r<BR>\n" );
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document.write( "the second column through the 10th column provide the hundredth portion of the z-score.\r<BR>\n" );
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document.write( "a z-score of -.4 is equal to -4.0 plus .00 column which is the area in the second column of that row.\r<BR>\n" );
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document.write( "a z-score of -.42 would be equal to a -4.0 plus .03 .02 column which would be the area in the 4th column of that row.\r<BR>\n" );
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document.write( "in numbers, -.4 gets you .3446 which is in the second column of that row and -.42 gets you .3372 which is in the fourth column of that row.\r<BR>\n" );
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document.write( "-.42 is equal to -.4 + .02 = -.42\r<BR>\n" );
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document.write( "the .01, .02, .03, ... etc columns are shown at the top of the table.\r<BR>\n" );
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document.write( "you can see that clearer in the second of the two displays of the z-score table.\r<BR>\n" );
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