document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=298823>Question 298823</A>: <I><SPAN STYLE=\"word-break: break-all;\">  1. By measuring the amount of time it takes a component of a product to move from one workstation to the next, an engineer has estimated that the standard deviation is 5.1 seconds. <BR>\n" );
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document.write( "Answer each of the following (show all work):<BR>\n" );
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document.write( "(A) How many measurements should be made in order to be 95% certain that the maximum error of estimation will not exceed 1.5 seconds? <BR>\n" );
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document.write( "(B) What sample size is required for a maximum error of 2.5 seconds? \r<BR>\n" );
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document.write( " 2. Determine the critical region and critical values for z that would be used to test the null hypothesis at the given level of significance, as described in each of the following:   <BR>\n" );
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document.write( "(A) Ho:&#61549; &#8804; 20 and Ha: &#61549; > 20,  &#945; = 0.05<BR>\n" );
document.write( "(B) Ho: &#61549; = 25and  Ha: &#61549; &#8800; 25,   &#945; = 0.01<BR>\n" );
document.write( "(C) Ho: &#61549; &#8805; 18and  Ha: &#61549; <  18,  &#945; = 0.01<BR>\n" );
document.write( "3. A 99% confidence interval estimate for a population mean was computed to be (42.0 to 50.0). Determine the mean of the sample, which was used to determine the interval estimate (show all work).<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #214903 by </B><A HREF=/tutors/aboutme.mpl?userid=alanc><B>alanc(27)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=alanc><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=alanc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=214903\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A.)  Ok, we know our confidence interval, it is a 95% interval.\r<BR>\n" );
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document.write( "Our variable of interest X : the amount of time a component moves from one station to another.<BR>\n" );
document.write( "s = 5.1 sec  (Standard deviation)<BR>\n" );
document.write( "We want to find n: the number of measurements to make such that our error doesn't exceed 1.5\r<BR>\n" );
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document.write( "I will assume this is a Normal distribution<BR>\n" );
document.write( "Error is X - u.   The Z variable is  z = (x-u)/[s/sqrt(n)]<BR>\n" );
document.write( "We want the Probability(X-u <= 1.5 ) >= 0.95\r<BR>\n" );
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document.write( "So:   P((x-u)/[s/sqrt(n)]  <= 1.5/[5.1*sqrt(n)] ) >= 0.95\r<BR>\n" );
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document.write( "But since the normal distribution is symmetric about mean, I will look for a z-value corresponding to 0.95/2 = 0.475, which is z= 1.96\r<BR>\n" );
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document.write( "from the z-score chart this corresponds to :  0 < z <= 1,96\r<BR>\n" );
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document.write( "Z = (x-u)/[s/sqrt(n)]  <= 1.5/[5.1*sqrt(n)]\r<BR>\n" );
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document.write( "therefore  sqrt(n)* 1.5/5.1 = 1.96\r<BR>\n" );
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document.write( "n = (5.1*1.96/1.5)^2<BR>\n" );
document.write( "n = 44.408\r<BR>\n" );
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document.write( "so I would say 44 or more tests.<BR>\n" );
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document.write( "The estimator for standard deviation  is  s/sqrt(n)  not s/sqrt(n -1)<BR>\n" );
document.write( " that is why I have Z = (x-u)/[s/sqrt(n)],  if I used s/sqrt(n-1) as my estimator, then Z = (x-u)/[s/sqrt(n - 1)] would be my Z statistic  and then n would be 45 or more tests.\r<BR>\n" );
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document.write( "I assumed we used a normal distribution because the number of measurements was large.  If the number of samples is small, we would be forced to use the t-distribution.\r<BR>\n" );
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