document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165659>Question 1165659</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The monthly earnings of a group of recent business school graduates are normally distributed with a standard deviation of 589 dollars. A researcher wants to estimate the mean monthly earnings of all recent graduates from the school. Find the sample size needed to have a confidence level of 95% and a margin of error of 132 dollars. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852952 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852952\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sample size needed to estimate the mean monthly earnings is **77**.\r<BR>\n" );
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document.write( "To find the required sample size ($n$), we use the formula for estimating a population mean:\r<BR>\n" );
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document.write( "$$n = \left(\frac{Z \cdot \sigma}{ME}\right)^2$$\r<BR>\n" );
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document.write( "### 1. Identify the Variables\r<BR>\n" );
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document.write( "* **Standard Deviation ($\sigma$):** $\$589$<BR>\n" );
document.write( "* **Margin of Error ($ME$):** $\$132$<BR>\n" );
document.write( "* **Confidence Level:** $95\%$<BR>\n" );
document.write( "* **Z-score ($Z$):** For a $95\%$ confidence level, the critical $Z$-score is $\mathbf{1.96}$. \r<BR>\n" );
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document.write( "### 2. Calculate the Sample Size\r<BR>\n" );
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document.write( "Substitute the values into the formula:\r<BR>\n" );
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document.write( "$$n = \left(\frac{1.96 \times 589}{132}\right)^2$$\r<BR>\n" );
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document.write( "First, calculate the term inside the parenthesis:\r<BR>\n" );
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document.write( "$$n = \left(\frac{1154.44}{132}\right)^2$$<BR>\n" );
document.write( "$$n \approx (8.745757...)^2$$<BR>\n" );
document.write( "$$n \approx 76.487$$\r<BR>\n" );
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document.write( "### 3. Round Up\r<BR>\n" );
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document.write( "Since the sample size must be a whole number and you must err on the side of caution (to ensure the margin of error is *no more* than \$132), you must **always round up** to the next whole number.\r<BR>\n" );
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document.write( "$$n = \mathbf{77}$$\r<BR>\n" );
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document.write( "The researcher needs a sample size of **77** graduates. </SPAN>\n" );
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