document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083698>Question 1083698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4. A light bulb manufacturer produces bulbs which have a mean life of 1400 hours and a standard deviation of 200 hours. Assuming a normal distribution.\r<BR>\n" );
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document.write( "(i) What proportion of scores would be expected to have a life of between 1600 and 1900 hours?<BR>\n" );
document.write( "(ii) What number of hours of life can the manufacturer guarantee so that there are no more than 2%<BR>\n" );
document.write( "rejects?<BR>\n" );
document.write( "(iii) What percentage of light bulbs would have a life of more than 1850 hours? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697808 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=697808\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "A light bulb manufacturer produces bulbs which have a mean life of 1400 hours and a standard deviation of 200 hours. Assuming a normal distribution. <BR>\n" );
document.write( "(i) What proportion of scores would be expected to have a life of between 1600 and 1900 hours?<BR>\n" );
document.write( "(ii) What number of hours of life can the manufacturer guarantee so that there are no more than 2%<BR>\n" );
document.write( "rejects?<BR>\n" );
document.write( "(iii) What percentage of light bulbs would have a life of more than 1850 hours?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution.<BR>\n" );
document.write( "All questions are based on the normal distribution with parameters N(0,1), i.e. mean, mu = 0, standard deviation, sigma = 1.<BR>\n" );
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document.write( "All data should be normalized to the distribution N(0,1) to use the normal distribution table.<BR>\n" );
document.write( "Normalized value is<BR>\n" );
document.write( "Z=(X-mean)/sigma=(X-1400)/200<BR>\n" );
document.write( "then P(X<=Z) can be found from the probability table.<BR>\n" );
document.write( "(for example, http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf)<BR>\n" );
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document.write( "(i) <BR>\n" );
document.write( "Z(1600 hours)=(1600-1400)/200=1<BR>\n" );
document.write( "Z(1900 hours)=(1900-1400)/200=2.5<BR>\n" );
document.write( "P(1600<= X <= 1900)=Z(1900)-Z(1600)=P(2.5)-P(1)=0.9938-0.8413=0.1524<BR>\n" );
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document.write( "(ii)<BR>\n" );
document.write( "Assume N hours will give an expected defective percentage of 0.02, then<BR>\n" );
document.write( "P(X<N)=0.98 => P(Z=2.054) from table.<BR>\n" );
document.write( "Therefore<BR>\n" );
document.write( "(N-1400)/200=2.054 => N=1400+200*2.054=1810.75 hours<BR>\n" );
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document.write( "(iii)<BR>\n" );
document.write( "X=1850, Z=(1850-1400)/200=2.25<BR>\n" );
document.write( "P(x<=X)=P(Z<2.25)=0.9878\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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