document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209758>Question 1209758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Provide a 96% confidence interval for 𝐼 = ∫₋∞^∞ e^(−x²)|sin x| dx. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850196 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53763)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850196\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "Strictly saying, the problem and the question do not make sense,\r<BR>\n" );
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document.write( "since the integral is some certain real number - not a random variable.\r<BR>\n" );
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document.write( "So, my diagnosis is that this \"problem\" is a kind of nonsense.\r<BR>\n" );
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document.write( "Soup of words and mathematical symbols with no sense.\r<BR>\n" );
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document.write( "In a normal human language, &nbsp;you want to find &nbsp;X &nbsp;in a way, &nbsp;that \r<BR>\n" );
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document.write( "for any &nbsp;x > X &nbsp;the integral of &nbsp;e(-t^2)|sin(t)| &nbsp;from &nbsp;x &nbsp;to infinity be less than &nbsp;(1-0.96)/2 = 0.02.\r<BR>\n" );
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document.write( "   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A physicist wouldn't bother.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;He would say: just replace  &nbsp;|sin(t)|  &nbsp;by  &nbsp;1 &nbsp;(one unit)  &nbsp;and look for &nbsp;X &nbsp;such\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;that for any &nbsp;x > X &nbsp;integral of  &nbsp;e(-t^2)  &nbsp;to infinity is less than &nbsp;0.02.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is because &nbsp; 1  &nbsp;\" majorizes \"  &nbsp;|sin(x)|.\r<BR>\n" );
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document.write( "Then you come to the  &nbsp;STANDARD  &nbsp;normal distribution function.\r<BR>\n" );
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document.write( "For it, &nbsp;you know from the &nbsp;\" empirical rule \", &nbsp;that the &nbsp;95% &nbsp;symmetric confidence interval <BR>\n" );
document.write( "is two standard deviations from the mean, &nbsp;which is zero in your case.\r<BR>\n" );
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document.write( "I think, &nbsp;that for you it is the same now, &nbsp;96% &nbsp;confidence or &nbsp;95% confidence.\r<BR>\n" );
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document.write( "So, &nbsp;with this rude approach, &nbsp;the value &nbsp;X = 2  &nbsp;(two standard deviations)  &nbsp;is what you need.\r<BR>\n" );
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document.write( "It is how a physicist or an engineer would solve/answer your question <BR>\n" );
document.write( "after 5 minutes thinking and without making complicated calculations.\r<BR>\n" );
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document.write( "Or any other person, having common sense in his mind.\r<BR>\n" );
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document.write( "Let me know if it makes sense to you.\r<BR>\n" );
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