document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120034>Question 1120034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 75 hours. A random sample of 36 light bulbs indicated a sample mean life of 260 hours. Complete parts&#8203; (a) through&#8203; (d) below.<BR>\n" );
document.write( "a. Construct a 95&#8203;% confidence interval estimate for the population mean life of light bulbs in this shipment.<BR>\n" );
document.write( "The 95&#8203;% confidence interval estimate is from a lower limit of <BR>\n" );
document.write( "nothing hours to an upper limit of <BR>\n" );
document.write( "nothing hours.<BR>\n" );
document.write( "&#8203;(Round to one decimal place as&#8203; needed.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735702 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=735702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the 95% confidence interval requires a critical z-score of plus or minus 1.959963986.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score formula is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z is the z-score.<BR>\n" );
document.write( "x is the raw score.<BR>\n" );
document.write( "m is the raw mean.<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your raw mean is 260 hours.<BR>\n" );
document.write( "your population standard deviation is 75 hours.<BR>\n" );
document.write( "your sample size is 36.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "formula for standard error is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "standard error = standard deviation / square root of size.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in your problem, you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s = 75 / sqrt(36) = 75 / 6 = 12.5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "use the critical z-scores and the standard error and the mean to find the critical raw scores.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "formula is z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z = plus or minus 1.959963986.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s = 12.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "on the low side, the z-score formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-1.959963986 = (x - 260) / 12.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x to get x = 235.5004502\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "on the high side, the z-score formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.959963986 = (x - 260) / 12.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x to get x = 284.4995498.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's your 95% confidence interval.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "following is a picture of the 95% confidence interval using z-scores.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/071815.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "following is a picture of the 95% confidence interval using raw scores.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/071816.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in both, the area under the normal distribution curve is shown as .95.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the same as 95%.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your confidence interval is the shaded area on the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the critical z-score was found using the inverse norm function of the TI-84 Plus calculator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );