document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186222>Question 1186222</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It has been estimated that 40% of the televisions that the manufacturer makes need repairs in the first three years of operation. A new hotel buys 90 televisions from the company. Find the probability that in the first three years operation:<BR>\n" );
document.write( "a.	Less than 32 of the television needs repairs<BR>\n" );
document.write( "b.	Between 38 and 42 of the televisions, needs repairs </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817176 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=817176\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this is a binomial (n, p) with n=90 and p=0.4<BR>\n" );
document.write( "with normal approximation with mean 36 and sd  (see below) of 4.65<BR>\n" );
document.write( "z <(31.5-36)/4.65=-0.968<BR>\n" );
document.write( "that probability is 0.1665<BR>\n" );
document.write( "from calculator (binomcdf(90,0.4,31)) which gives everything 31 and fewer is 0.1666\r<BR>\n" );
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document.write( "second can be done using formula where <BR>\n" );
document.write( "px=38) is 90C38*0.4^38*0.6^52=0.0774<BR>\n" );
document.write( "px=39) is 0.0688 (note the mean is 90*0.4=36, so after that, the probability should fall slightly, which it is.<BR>\n" );
document.write( "p(x=40) is 0.0585<BR>\n" );
document.write( "p(x=41)= is 0.0476<BR>\n" );
document.write( "p(x=42) is 0.0370\r<BR>\n" );
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document.write( "That sum 0.2893, which is exact.<BR>\n" );
document.write( "-<BR>\n" );
document.write( "check with normal approximation<BR>\n" );
document.write( "mean=np=36<BR>\n" );
document.write( "variance is np(1-p)=36*0.6=21.6<BR>\n" );
document.write( "sort (V)=sd=4.65<BR>\n" );
document.write( "so between 38 and 42<BR>\n" );
document.write( "z=(42.5-36)/4.65, using continuity correction factor<BR>\n" );
document.write( "=1.40<BR>\n" );
document.write( "and <BR>\n" );
document.write( "z=(37.5-36)/4.65<BR>\n" );
document.write( "=0.323<BR>\n" );
document.write( "probability is in that z interval which is 0.2925, which is a decent approximation. </SPAN>\n" );
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