document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083276>Question 1083276</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A donut baker wants to ensure that the radius of its donuts is within +- 0.25cm. Based on past statistical studies, she knows that the donut’s radius has a standard deviation of 1.25cm. How many measurements should she take to ensure a 90 percent confidence in the radius of the donuts are within 0.25cm, rather than measure each and every donut that is baked? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697344 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=697344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think it works this way.\r<BR>\n" );
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document.write( "at 90% confidence level, your critical z-score will be plus or minus 1.644853626.\r<BR>\n" );
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document.write( "i used the ti-84 plus calculator to find this.<BR>\n" );
document.write( "90% confidence level using a 2-tailed distribution leaves an alpha of.05 on each end.<BR>\n" );
document.write( "the plus z-score gives you an alpha of .05 on the right end of the distribution.<BR>\n" );
document.write( "the minus z-score gives you an alpha of.05 on the left end of the distribution.<BR>\n" );
document.write( "add up both tails and you get an overall alpha of .10 which give you 90% of the normal distribution within the middle.\r<BR>\n" );
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document.write( "the formula for sample z-score is z = (x-m)/se\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( "se is the standard error\r<BR>\n" );
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document.write( "the standard error is calculated by the formula of se = sd / sqrt(n)\r<BR>\n" );
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document.write( "sd is the standard deviation<BR>\n" );
document.write( "n is the sample size\r<BR>\n" );
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document.write( "if you replace se with sd / sqrt(n), then the formula for z-score becomes z = (x-m) / (sd / sqrt(n))\r<BR>\n" );
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document.write( "this is equivalent to z = (x-m) * sqrt(n)) / sd\r<BR>\n" );
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document.write( "at the limits, you want (x-m) to be equal to plus or minus .25.\r<BR>\n" );
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document.write( "therefore, the formula will become:\r<BR>\n" );
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document.write( "1.644853626 = .25 * sqrt(n) / 1.25\r<BR>\n" );
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document.write( "that's because the critical z-score is 1.644853626 and the standard deviation is 1.25\r<BR>\n" );
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document.write( "solve for sqrt(n) to get sqrt(n) = 1.644853626 / .25 * 1.25 = 8.22426813\r<BR>\n" );
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document.write( "solve for n to get n = 67.63\r<BR>\n" );
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document.write( "you would round this up to 68.\r<BR>\n" );
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document.write( "the sample size should be at least 68 in order to ensure that the limits are within plus or minus .25 at 90% confidence level.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "assume the sample mean is 10 cm.\r<BR>\n" );
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document.write( "the lower limit would have to be 9.75 cm and the upper limit would have to be 10.25 cm.\r<BR>\n" );
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document.write( "your standard deviation is 1.25\r<BR>\n" );
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document.write( "your standard error is 1.25 / sqrt(68) = .1525847656\r<BR>\n" );
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document.write( "your lower z-score would be (9.75-10)/.1525847656 = -1.64924225\r<BR>\n" );
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document.write( "your upper z-score would be (10.25-10)/.1525847656 = 1.64924225\r<BR>\n" );
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document.write( "a z-score of-1.64924225 gives you an alpha of .0495489908 on the left end of the distribution curve.\r<BR>\n" );
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document.write( "a z-score of 1.64924225 gives you an alpha of .0495489908 on the right end of the distribution curve.\r<BR>\n" );
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document.write( "that's a 90% confidence interval.\r<BR>\n" );
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document.write( "90% of your sample means will be between these limits.\r<BR>\n" );
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document.write( "if you had rounded the critical z-score to plus or minus 1.645, you would have gotten the same result.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "1.645 = .25/1.25 * sqrt(n)\r<BR>\n" );
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document.write( "sqrt(n) = 1.645 * 1.25 / .25 = 8.225\r<BR>\n" );
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document.write( "8.225^2 = 67.670625 which rounds up to 68.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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