document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175995>Question 1175995</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following is a frequency distribution of ages of residents of a senior citizens' home in New York. Fill in the midpoints, then calculate the mean and the standard deviation. Round to two decimal positions. You may assume the data represents a population.\r<BR>\n" );
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document.write( "Ages of Residents:59 - 65, 66 - 72, 73 - 79, 80 - 86<BR>\n" );
document.write( "Midpoints:62,69,76,83<BR>\n" );
document.write( "Frequency:2,3,4,1\r<BR>\n" );
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document.write( "Mean (μ)= <BR>\n" );
document.write( "Standard deviation (σ) = <BR>\n" );
document.write( "Round the standard deviation up to 2 decimal places if necessary. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850599 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the mean and standard deviation for the given frequency distribution.\r<BR>\n" );
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document.write( "**1. Calculate the Mean (μ)**\r<BR>\n" );
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document.write( "* We use the formula: μ = Σ(midpoint * frequency) / Σfrequency<BR>\n" );
document.write( "* Σ(midpoint * frequency) = (62 * 2) + (69 * 3) + (76 * 4) + (83 * 1) = 124 + 207 + 304 + 83 = 718<BR>\n" );
document.write( "* Σfrequency = 2 + 3 + 4 + 1 = 10<BR>\n" );
document.write( "* μ = 718 / 10 = 71.80\r<BR>\n" );
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document.write( "**2. Calculate the Standard Deviation (σ)**\r<BR>\n" );
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document.write( "* We use the formula: σ = √[Σ(frequency * (midpoint - μ)²) / Σfrequency]<BR>\n" );
document.write( "* First, calculate (midpoint - μ)² for each midpoint:<BR>\n" );
document.write( "    * (62 - 71.8)² = (-9.8)² = 96.04<BR>\n" );
document.write( "    * (69 - 71.8)² = (-2.8)² = 7.84<BR>\n" );
document.write( "    * (76 - 71.8)² = (4.2)² = 17.64<BR>\n" );
document.write( "    * (83 - 71.8)² = (11.2)² = 125.44<BR>\n" );
document.write( "* Next, multiply each result by its corresponding frequency:<BR>\n" );
document.write( "    * 96.04 * 2 = 192.08<BR>\n" );
document.write( "    * 7.84 * 3 = 23.52<BR>\n" );
document.write( "    * 17.64 * 4 = 70.56<BR>\n" );
document.write( "    * 125.44 * 1 = 125.44<BR>\n" );
document.write( "* Sum these values: 192.08 + 23.52 + 70.56 + 125.44 = 411.6<BR>\n" );
document.write( "* Divide by the total frequency: 411.6 / 10 = 41.16<BR>\n" );
document.write( "* Take the square root: √41.16 ≈ 6.415606\r<BR>\n" );
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document.write( "* Rounding to two decimal places: σ ≈ 6.42\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "* Mean (μ) = 71.80<BR>\n" );
document.write( "* Standard Deviation (σ) = 6.42<BR>\n" );
document.write( " </SPAN>\n" );
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