document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=153491>Question 153491</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were<BR>\n" );
document.write( "3.087	3.131	3.241	3.241	3.270	3.353	3.400	3.411	3.437	3.477<BR>\n" );
document.write( "a). Construct a 90 percent confidence interval for the true mean weight. (b) What sample size would be necessary to estimate the true weight with an error of ± 0.03 grams with 90 percent confidence? (c) Discuss the factors which might cause variation in the weight of Tootsie Rolls during manufacture. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #112997 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=112997\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)Since you don't know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > and your sample size is <30, assume your samples are  normally distributed. <BR>\n" );
document.write( "Use the t-distribution as the best estimate. Degrees of freedom are n-1 or 9. <BR>\n" );
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document.write( "The calculated mean,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bm%5D ALIGN=MIDDLE  ALT=\"x%5Bm%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is 3.3048.<BR>\n" );
document.write( "The calculated standard deviation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s ALIGN=MIDDLE  ALT=\"s\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is 0.132<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=0.10 ALIGN=MIDDLE  ALT=\"alpha=0.10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DOF=9 ALIGN=MIDDLE  ALT=\"DOF=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1.833 ALIGN=MIDDLE  ALT=\"t=1.833\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "The confidence interval is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bm%5D-t%28s%2Fsqrt%28n%29%29%3C+mu+%3C+x%5Bm%5D%2Bt%28s%2Fsqrt%28n%29%29 ALIGN=MIDDLE  ALT=\"x%5Bm%5D-t%28s%2Fsqrt%28n%29%29%3C+mu+%3C+x%5Bm%5D%2Bt%28s%2Fsqrt%28n%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.3048-1.833%280.132%2Fsqrt%2810%29%29%3C+mu+%3C+3.3048%2B1.833%280.132%2Fsqrt%2810%29%29 ALIGN=MIDDLE  ALT=\"3.3048-1.833%280.132%2Fsqrt%2810%29%29%3C+mu+%3C+3.3048%2B1.833%280.132%2Fsqrt%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.228%3Cmu+%3C3.381 ALIGN=MIDDLE  ALT=\"3.228%3Cmu+%3C3.381\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "b) Working backwards,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%28s%2Fsqrt%28n%29%29=0.03 ALIGN=MIDDLE  ALT=\"t%28s%2Fsqrt%28n%29%29=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%280.132%2Fsqrt%28n%29%29=0.03 ALIGN=MIDDLE  ALT=\"t%280.132%2Fsqrt%28n%29%29=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2Fsqrt%28n%29=+0.227 ALIGN=MIDDLE  ALT=\"t%2Fsqrt%28n%29=+0.227\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "t is a function of n so you have to iterate to find n.<BR>\n" );
document.write( "I set up an iteration cell in EXCEL using TINV and varying n.<BR>\n" );
document.write( "n=54 gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2Fsqrt%28n%29=+0.227743 ALIGN=MIDDLE  ALT=\"t%2Fsqrt%28n%29=+0.227743\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "54 samples required to give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%2B-+0.03 ALIGN=MIDDLE  ALT=\"0+%2B-+0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For this large a value for n, we can use the normal distribution as a check of the value since as n gets large the t distribution approaches the normal distribution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%28%28z%2A%28sigma%29%29%2F0.03%29%5E2 ALIGN=MIDDLE  ALT=\"n=%28%28z%2A%28sigma%29%29%2F0.03%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "z=1.65 for 90%<BR>\n" );
document.write( "Use the calculated 0.132 as an estimate for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%28%281.65%2A%280.132%29%29%2F0.03%29%5E2 ALIGN=MIDDLE  ALT=\"n=%28%281.65%2A%280.132%29%29%2F0.03%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%287.26%29%5E2 ALIGN=MIDDLE  ALT=\"n=%287.26%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=52.7 ALIGN=MIDDLE  ALT=\"n=52.7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=53 ALIGN=MIDDLE  ALT=\"n=53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Good, that's close.\r<BR>\n" );
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document.write( "c) Factors include length errors from cutting machine, air in the mixture, humidity in the plant, density changes in the mixture, etc.  </SPAN>\n" );
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