document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1081974>Question 1081974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of the number of farms (in thousands) in various states follows. Estimate the mean number of farms per state with 99% confidence. Assume o=31. Round intermediate and final answers to one decimal place. Assume the population is normally distributed.<BR>\n" );
document.write( "47    50    40    109    78   6   52    21    15    7    68<BR>\n" );
document.write( " 16     48   79   44\r<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 #696151 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=696151\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> With 99% confidence, the z critical value is approximately 2.576\r<BR>\n" );
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document.write( "Use a calculator or a <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">table</a> to determine this. If you decide to use the table that I've linked, look at the \"99%\" in the bottom row. Then look at the value 2.576 just above it.\r<BR>\n" );
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document.write( "So we know that z = 2.576\r<BR>\n" );
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document.write( "Now let's compute the sample mean (xbar) of the given data set. Simply add up the values and divide by 15. We divide by 15 because there are 15 values in the data set\r<BR>\n" );
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document.write( "Add up the values: 47+50+40+109+78+6+52+21+15+7+68+16+48+79+44 = 680\r<BR>\n" );
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document.write( "Divide the sum by 15<BR>\n" );
document.write( "680/15 = 45.333333<BR>\n" );
document.write( "Approximate to 6 decimal places\r<BR>\n" );
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document.write( "So xbar = 45.333333 roughly\r<BR>\n" );
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document.write( "We'll use these formulas\r<BR>\n" );
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document.write( "L = xbar - z*sigma/sqrt(n)<BR>\n" );
document.write( "U = xbar + z*sigma/sqrt(n)\r<BR>\n" );
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document.write( "Where<BR>\n" );
document.write( "xbar = sample mean<BR>\n" );
document.write( "z = critical value<BR>\n" );
document.write( "sigma = population standard deviation<BR>\n" );
document.write( "n = sample size<BR>\n" );
document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "U = upper bound of confidence interval\r<BR>\n" );
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document.write( "In this case,<BR>\n" );
document.write( "xbar = 45.333333 (approximate; see above)<BR>\n" );
document.write( "z = 2.576 (approximate; see above)<BR>\n" );
document.write( "sigma = 31 (given in the instructions)<BR>\n" );
document.write( "n = 15 (since there are fifteen values given)<BR>\n" );
document.write( "L = unknown value<BR>\n" );
document.write( "U = unknown value\r<BR>\n" );
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document.write( "The values L and U are unknown for now, but we can use the other values to find L and U.\r<BR>\n" );
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document.write( "Let's compute the lower boundary L<BR>\n" );
document.write( "L = xbar - z*sigma/sqrt(n)<BR>\n" );
document.write( "L = 45.333333 - 2.576*31/sqrt(15)<BR>\n" );
document.write( "L = 24.714602<BR>\n" );
document.write( "L = 24.7\r<BR>\n" );
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document.write( "Let's compute the upper boundary U<BR>\n" );
document.write( "U = xbar + z*sigma/sqrt(n)<BR>\n" );
document.write( "U = 45.333333 + 2.576*31/sqrt(15)<BR>\n" );
document.write( "U = 65.9520635396493<BR>\n" );
document.write( "U = 66.0\r<BR>\n" );
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document.write( "After rounding to 1 decimal place, the 99% confidence interval is (L, U) = (24.7, 66.0)\r<BR>\n" );
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document.write( "We're 99% confident that the mean (mu) is between 24.7 and 66.0\r<BR>\n" );
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document.write( "Answer: <font color=red>(24.7,66.0)</font> which is a confidence interval (not an ordered pair). </SPAN>\n" );
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