document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183927>Question 1183927</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am stuck on how to solve this. Use the given margin of error, confidence level, and population standard deviation  σ  to find the minimum sample size required to estimate an unknown population mean  μ.\r<BR>\n" );
document.write( "\n" );
document.write( "1. Margin of error: 1 inch; confidence level: 90%;  σ  = 2.5 inches\r<BR>\n" );
document.write( "\n" );
document.write( "2. Margin of error: 0.5 inches; confidence level: 90%; σ = 2.4 inches </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814440 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=814440\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i usually calculate it this way.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula to use is z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you want the  margin of error to be equal to 1, then (x - m) must be equal to 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z = 1 / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "at 90% confidence level, the two tailed condidence interval will require a cr5iticql z-score of plus or minus 1.645.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i usually work with the positive z-score to find the sample size required.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes 1.645 = 1 / s.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s is equal to the population standard deviation divided by the square root of the sample size.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s = 2.5 / sqrt(n), where n is the sample size.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 = 1 / (2.5 / sqrt(n)).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since 1 / (2.5 / sqrt(n)) is equal to 1 / 2.5 * sqrt(n), the equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 = 1 / 2.5 * sqrt(n).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of this equaton by 2.5 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 * 2.5 = sqrt(n).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for sqrt(n) to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(n) = 4.1125.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for n to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n = 16.91265625.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's your solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since s is equal to 2.5 / sqrt(n), then s is equal to 2.5 / 4.1125 which makes s = .607903 rounded to 6 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this value of the standard error = s will give you the margin of error you desire.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this margin of error will be the same, regardless of what the mean is.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "some samples are shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082701.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082702.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082703.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can see that the margin of error is plus or minus 1 regardless of what the mean is.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for your second problem, i did the same thing.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "critical z was still 1.645 because the confidence level was still 90%.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the margin of eror desired is not plus or minus .5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the population standard deviation is now 2.4, rather than 2.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score formula ia still z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you want a margin of error of .5, then (x - m) must be equal to .5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 = .5 / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of the formula by s to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 * s = .5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since s = 2.4 / sqrt(n), the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.645 * 2.4 / sqrt(n) = .5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for sqrt(n) to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrtn) = 1.645 * 2.4 / .5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes sqrt(n) = 7.896\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for n to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n = 7.896^2 = 62.346816.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when sqrt(n) = 7.896, then s = 2.4 / 7.896 = .3039513 rounded to 6 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is the standard error that will give you a margin of error of plus or minus .5 regardless of what the mean is.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "some samples are shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082704.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082705.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/082706.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i believe you can generalize the formula as shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score formula is z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when (x - m) is the MOE, the formula becdomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z = MOE / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of the formula by s to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z * s = MOE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since s = population standard deviation / sqrt(n), the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z * population standard deviation / sqrt(n) = MOE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for sqrt(n) to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(n) = z * population standard deviation / MOE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let's see if this works.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when z = 1.645 and MOE = 1 and population standard deviation = 2.5, the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(n) = 1.645 * 2.5 / 1 = 4.1125.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this agrees with what we got before.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when z = 1.645 and MOE = .5 and population standard deviation = 2.4, the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(n) = 1.645 * 2.4 / .5 = 7.896.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this also agrees with what we got before.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score used is the critical z-score for the confidence level indicated.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );