document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084646>Question 1084646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Statistics calculated from a sample of 22 observations are: \r<BR>\n" );
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document.write( "&#8721; (n=22 and i=1) x = 1451\r<BR>\n" );
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document.write( "&#8721; (n=22 and i=1) x^2 = 106639\r<BR>\n" );
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document.write( "(a) Find the sample mean: I found this to be 1451/22<BR>\n" );
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document.write( "(b) What is the sample standard deviation? <BR>\n" );
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document.write( "(c) Assume that the population distribution is normal. Find a 95% confidence interval for the population mean. (    ,     ) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698717 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=698717\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"4\">Part (a) <BR>\n" );
document.write( "Correct. The fraction 1451/22 approximates to 65.9545454545454<BR>\n" );
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document.write( "Part (b)<BR>\n" );
document.write( "Notation notes:<BR>\n" );
document.write( "When I say \"sigma(X^2)\" I mean <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sum X^2\"> which in this case is 106639 <BR>\n" );
document.write( "Similarly, \"sigma(X)\" means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sum X\"> which in this case is 1451\r<BR>\n" );
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document.write( "Using those values and n = 22, we can use the formula below to get\r<BR>\n" );
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document.write( "s = sqrt( (n*sigma(X^2)-(sigma(X))^2)/(n*(n-1)) )<BR>\n" );
document.write( "s = sqrt( (22*106639-(1451)^2)/(22*(22-1)) )<BR>\n" );
document.write( "s = sqrt( (22*106639-2105401)/(22*21) )<BR>\n" );
document.write( "s = sqrt( (2346058-2105401)/(462) )<BR>\n" );
document.write( "s = sqrt( (240657)/(462) )<BR>\n" );
document.write( "s = sqrt( 520.902597402597 )<BR>\n" );
document.write( "s = <font color=red>22.8232906786598</font>\r<BR>\n" );
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document.write( "which is the approximate answer to part (b)<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "We're given n = 22 as the sample size.\r<BR>\n" );
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document.write( "Because n > 30 is not true and we don't know sigma (population standard deviation) we must use the T distribution.<BR>\n" );
document.write( "The degrees of freedom (df) are<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 22-1<BR>\n" );
document.write( "df = 21\r<BR>\n" );
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document.write( "The value of xbar is approximately 65.9545454545454 as done in part (a)\r<BR>\n" );
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document.write( "Use a table <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">such as this one</a> to locate the df = 21 row. <BR>\n" );
document.write( "I have marked this row with a red rectangle (see below)<BR>\n" );
document.write( "In this df = 21 row, mark the value that is directly over top the \"95%\" confidence level<BR>\n" );
document.write( "I marked this entire column with a blue rectangle (see below)<BR>\n" );
document.write( "The red and blue rectangles intersect at the value 2.080<BR>\n" );
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document.write( "So the t critical value is t = 2.080\r<BR>\n" );
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document.write( "Earlier in part (b) we calculated the sample standard deviation to be approximately s = 22.8232906786598\r<BR>\n" );
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document.write( "We take these values (xbar = 65.9545454545454, t = 2.080, s = 22.8232906786598, n = 22) to plug into the formulas below\r<BR>\n" );
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document.write( "L = xbar - t*(s/sqrt(n))<BR>\n" );
document.write( "L = 65.9545454545454 - 2.08*(22.8232906786598/sqrt(22))<BR>\n" );
document.write( "L = 65.9545454545454 - 2.08*(22.8232906786598/4.69041575982343)<BR>\n" );
document.write( "L = 65.9545454545454 - 2.08*(4.8659419222826)<BR>\n" );
document.write( "L = 65.9545454545454 - 10.1211591983478<BR>\n" );
document.write( "L = 55.8333862561976\r<BR>\n" );
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document.write( "U = xbar + t*(s/sqrt(n))<BR>\n" );
document.write( "U = 65.9545454545454 + 2.08*(22.8232906786598/sqrt(22))<BR>\n" );
document.write( "U = 65.9545454545454 + 2.08*(22.8232906786598/4.69041575982343)<BR>\n" );
document.write( "U = 65.9545454545454 + 2.08*(4.8659419222826)<BR>\n" );
document.write( "U = 65.9545454545454 + 10.1211591983478<BR>\n" );
document.write( "U = 76.0757046528932\r<BR>\n" );
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document.write( "The 95% confidence interval for the mean (mu) is <font color=red>(L,U) = (55.8333862561976,76.0757046528932)</font>, which is an approximate interval.<BR>\n" );
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