document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175141>Question 1175141</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You intend to estimate a population mean μ from the following sample.<BR>\n" );
document.write( "51.7	41.4	56.8	74.3<BR>\n" );
document.write( "82	101.6	78	104.9<BR>\n" );
document.write( "55.5	77.3	72.6	72.2<BR>\n" );
document.write( "63.5	82.6	66.2	68.2<BR>\n" );
document.write( "79.5	64.5	56.4	55.8<BR>\n" );
document.write( "64.7	63	43.6	35.4<BR>\n" );
document.write( "82.7	59.8	89.7	83.5<BR>\n" );
document.write( "71.2	48.8	64	62.4<BR>\n" );
document.write( "83.7	77	64.7	60.6<BR>\n" );
document.write( "52.7	78.8	59.5	74.2<BR>\n" );
document.write( "65.5	83.8	43.7	63.4<BR>\n" );
document.write( "74	32	93.1	57.8<BR>\n" );
document.write( "86.6			\r<BR>\n" );
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document.write( "Find the 90% confidence interval. Enter your answer as a tri-linear inequality accurate to two decimal place (because the sample data are reported accurate to one decimal place) ... <μ< ...\r<BR>\n" );
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document.write( "2. You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ = 36.7. You would like to be 90% confident that your esimate is within 0.1 of the true population mean. How large of a sample size is required? n = Do not round mid-calculation. However, use a critical value accurate to three decimal places — this is important for the system to be able to give hints for incorrect answers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800743 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=800743\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1<BR>\n" );
document.write( "Use a spreadsheet or calculator to compute the following<BR>\n" );
document.write( "xbar = 67.8551020408163<BR>\n" );
document.write( "s = 15.8895151649404\r<BR>\n" );
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document.write( "xbar is the sample mean, while s is the sample standard deviation<BR>\n" );
document.write( "Because we have n = 49 items here, this means we have enough to use the Z distribution. We can use this if we know sigma, or if n > 30. When n > 30, the student T distribution is approximately the same as the standard normal distribution.\r<BR>\n" );
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document.write( "At 90% confidence, the z critical value is roughly z = 1.645; use a table or calculator to compute this\r<BR>\n" );
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document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "L = xbar - z*s/sqrt(n)<BR>\n" );
document.write( "L = 67.8551020408163 - 1.645*15.8895151649404/sqrt(49)<BR>\n" );
document.write( "L = 67.8551020408163 - 3.734036063761<BR>\n" );
document.write( "L = 64.1210659770552<BR>\n" );
document.write( "L = 64.12\r<BR>\n" );
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document.write( "U = upper bound of confidence interval<BR>\n" );
document.write( "U = xbar + z*s/sqrt(n)<BR>\n" );
document.write( "U = 67.8551020408163 + 1.645*15.8895151649404/sqrt(49)<BR>\n" );
document.write( "U = 67.8551020408163 + 3.734036063761<BR>\n" );
document.write( "U = 71.5891381045772<BR>\n" );
document.write( "U = 71.59\r<BR>\n" );
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document.write( "The 90% confidence interval of the form L < mu < U is therefore <font color=red>64.12 < mu < 71.59</font>\r<BR>\n" );
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document.write( "Problem 2\r<BR>\n" );
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document.write( "z = 1.645 from earlier (critical value for 90% confidence interval) <BR>\n" );
document.write( "sigma = 36.7 = given population standard deviation<BR>\n" );
document.write( "E = 0.1 = desired error\r<BR>\n" );
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document.write( "n = minimum sample size needed<BR>\n" );
document.write( "n = (z*sigma/E)^2<BR>\n" );
document.write( "n = (1.645*36.7/0.1)^2<BR>\n" );
document.write( "n = 364,471.801225<BR>\n" );
document.write( "n = <font color=red>364,472</font><BR>\n" );
document.write( "Always round up to the nearest whole number for this type of problem.\r<BR>\n" );
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document.write( "The min sample size needed is <font color=red>364,472</font>\r<BR>\n" );
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document.write( "Side note: you may need to ignore the comma if you are typing this answer into a computer system<BR>\n" );
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