document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1082200>Question 1082200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> First semester GPAs for a random selection of freshmen at a large university are shown. Estimate the true mean GPA of the freshmen class with 99% confidence. Assume o=0.62. Round intermediate and final answers to two decimal places. Assume the population is normally distributed. \r<BR>\n" );
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document.write( "3.1   3.8  3.0  3.2  2.8  2.7  2.5  1.9   2.0<BR>\n" );
document.write( "2.7  3.8  3.0  2.8  3.3  2.7  2.9  2.0  3.2 <BR>\n" );
document.write( "1.9  2.8  2.2  4.0  1.9  2.8  2.0  2.7  3.9\r<BR>\n" );
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document.write( "__________ < u < _________<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 #696261 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=696261\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given Data Set<BR>\n" );
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document.write( "	3.1,3.8,3.0,3.2,2.8,2.7,2.5,1.9,2.0,\r\n" );
document.write( "	2.7,3.8,3.0,2.8,3.3,2.7,2.9,2.0,3.2,\r\n" );
document.write( "	1.9,2.8,2.2,4.0,1.9,2.8,2.0,2.7,3.9\r\n" );
document.write( "	}\r\n" );
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document.write( "There are n = 27 values in the data set above (3 rows of 9 each)\r<BR>\n" );
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document.write( "Normally since n > 30 is not true, this means we'd have to use a T distribution. However, we're given the value of sigma (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%29 ALIGN=MIDDLE  ALT=\"sigma%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) so we can use the standard normal distribution. \r<BR>\n" );
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document.write( "Use a stats table to find the z critical value\r<BR>\n" );
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document.write( "I'm using <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">this table</a> specifically. Locate the \"99%\" value in the bottom row that corresponds to confidence level. The value just above that is  2.576\r<BR>\n" );
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document.write( "The 99% z critial value is z =  2.576\r<BR>\n" );
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document.write( "Sample mean\r<BR>\n" );
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document.write( "Add up all the values to get<BR>\n" );
document.write( "3.1+3.8+3.0+3.2+2.8+2.7+2.5+1.9+2.0+2.7+3.8+3.0+2.8+3.3+2.7+2.9+2.0+3.2+1.9+2.8+2.2+4.0+1.9+2.8+2.0+2.7+3.9 = 75.6\r<BR>\n" );
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document.write( "Divide this sum by n = 27<BR>\n" );
document.write( "75.6/n = 75.6/27 = 2.8\r<BR>\n" );
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document.write( "The sample mean is xbar = 2.8\r<BR>\n" );
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document.write( "So we have,<BR>\n" );
document.write( "xbar = 2.8 (computed above)<BR>\n" );
document.write( "z = 2.576 (approximate; see above)<BR>\n" );
document.write( "sigma = 0.62 (given)<BR>\n" );
document.write( "n = 27 \r<BR>\n" );
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document.write( "The lower boundary (L) of the confidence interval is<BR>\n" );
document.write( "L = xbar - z*(sigma/sqrt(n))<BR>\n" );
document.write( "L = 2.8 - 2.576*(0.62/sqrt(27))<BR>\n" );
document.write( "L = 2.49263411269062<BR>\n" );
document.write( "L = 2.49\r<BR>\n" );
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document.write( "The upper boundary (U) of the confidence interval is<BR>\n" );
document.write( "U = xbar + z*(sigma/sqrt(n))<BR>\n" );
document.write( "U = 2.8 + 2.576*(0.62/sqrt(27))<BR>\n" );
document.write( "U = 3.10736588730938<BR>\n" );
document.write( "U = 3.11\r<BR>\n" );
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document.write( "Overall the 99% confidence interval is (L, U) = (2.49, 3.11)\r<BR>\n" );
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document.write( "We have 99% confidence that the mean (mu) is between 2.49 and 3.11.\r<BR>\n" );
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document.write( "Answer: <font color=red>2.49</font> < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" > < <font color=red>3.11</font> </SPAN>\n" );
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