document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141192>Question 1141192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A travel agent wants to estimate the mean price of a flight from Boston to Denver.  A different airline study show that the standard deviation of flight tickets is $56.92.  The agent wants her estimate to be within $25 of the true mean flight cost.  How many flights must she include in her sample in order to be 90% in her estimate? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #761741 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=761741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the critical z-score at 90% confidence level is plus or minus 1.645.\r<BR>\n" );
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document.write( "the z-score formula is z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score.<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "the formula for s is:\r<BR>\n" );
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document.write( "s = standard deviation / square root of sample size = sd / sqrt(n).\r<BR>\n" );
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document.write( "for the score to be within 25 of the mean, then (x - m) must be equal to plus or minus 25.\r<BR>\n" );
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document.write( "we'll work the plus side.\r<BR>\n" );
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document.write( "the minus side should give you the same answer.\r<BR>\n" );
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document.write( "z = (x - m) / s becomes 1.645 = 25 / s which becomes 25 / (56.92/sqrt(n))\r<BR>\n" );
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document.write( "simplify to get 1.645 = 25 * sqrt(n) / 56.92\r<BR>\n" );
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document.write( "solve for sqrt(n) to get sqrt(n) = 1.645 * 56.92 / 25 = 3.745336.\r<BR>\n" );
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document.write( "solve for n to get n = 3.745336 ^ 2 = 14.02754175.\r<BR>\n" );
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document.write( "when the sample size is 14.02754175, s = sd / sqrt(n) becomes s = 56.92 / sqrt(14.02754175) = 15.19756839.\r<BR>\n" );
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document.write( "when z = 1.645, the z-score formula of z = (x - m) / s becomes 1.645 = (x - m) / 15.19756839.\r<BR>\n" );
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document.write( "solve for (x - m) to get (x - m) = 1.645 * 15.19756839) = 25.\r<BR>\n" );
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document.write( "if the z-score is -1.645, then z = (x - m) / s becomes -1.645 = (x - m) / 15.19756839 and you get (x - m) = -1.645 * 15.19756839 = -25.\r<BR>\n" );
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document.write( "when x - m = 25, then x = m + 25.\r<BR>\n" );
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document.write( "when x - m = -25, then x = m - 25.\r<BR>\n" );
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document.write( "your score will be within 25 dollars of the mean.\r<BR>\n" );
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document.write( "the mean could be anything and the formula will get you within 25 dollars of the mean.\r<BR>\n" );
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document.write( "for example, let the mean be 100 dollars.\r<BR>\n" );
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document.write( "at the 90% confidence level, the z-score is plus or minus 1.645.\r<BR>\n" );
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document.write( "the positive z-score formula becomes 1.645 = (x - 100) / 15.19756839.\r<BR>\n" );
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document.write( "solve for x to get x = 1.645 * 15.19756839 + 100 = 125.\r<BR>\n" );
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document.write( "the negative z-score formula becomes -1.645 = (x - 100) / 15.19756839.\r<BR>\n" );
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document.write( "solve for x to get x = -1.645 * 15.19756839 + 100 = 75.\r<BR>\n" );
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document.write( "125 is 25 dollars from the mean of 100.<BR>\n" );
document.write( "75 is 25 dollars from the mean of 100.\r<BR>\n" );
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document.write( "you can substitute any other mean and you will get the same distance from the mean, i.e. 25 dollars plus or minus from the mean.\r<BR>\n" );
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document.write( "to be exact, the sample size needed to be 14.02754175.\r<BR>\n" );
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document.write( "however, the number of flights is not a fraction, therefore you would go to the next higher integer.\r<BR>\n" );
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document.write( "she must include at least 15 flights to be 25 dollars or less from the true mean.\r<BR>\n" );
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document.write( "the following excel spreadsheet shows some of the figures from 10 to 20 flights in the sample size, including the one where we calculated 14.02754175.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/051832.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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