document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178061>Question 1178061</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How large should the size of a random sample be so that we can be 90% certain that the sample mean X will not deviate from the true mean by more than σ/2?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850368 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=850368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using Chebyshev's Inequality.\r<BR>\n" );
document.write( "\n" );
document.write( "**Understanding the Problem**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Confidence Level:** 90% (0.90)<BR>\n" );
document.write( "* **Deviation:** The sample mean (X̄) should not deviate from the true mean (μ) by more than σ/2.<BR>\n" );
document.write( "* **Goal:** Find the sample size (n).\r<BR>\n" );
document.write( "\n" );
document.write( "**Chebyshev's Inequality**\r<BR>\n" );
document.write( "\n" );
document.write( "Chebyshev's Inequality states:\r<BR>\n" );
document.write( "\n" );
document.write( "* P(|X̄ - μ| ≥ kσ_X̄) ≤ 1/k²<BR>\n" );
document.write( "* Or, P(|X̄ - μ| < kσ_X̄) ≥ 1 - 1/k²\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* X̄ is the sample mean<BR>\n" );
document.write( "* μ is the true mean<BR>\n" );
document.write( "* σ_X̄ is the standard deviation of the sample mean (σ/√n)<BR>\n" );
document.write( "* k is a positive constant\r<BR>\n" );
document.write( "\n" );
document.write( "**Applying Chebyshev's Inequality**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Set Up the Inequality:**<BR>\n" );
document.write( "    * We want P(|X̄ - μ| ≤ σ/2) ≥ 0.90.<BR>\n" );
document.write( "    * This is equivalent to P(|X̄ - μ| < σ/2) ≥ 0.90.<BR>\n" );
document.write( "    * We know that σ_X̄ = σ/√n.<BR>\n" );
document.write( "    * So, we have P(|X̄ - μ| < kσ/√n) ≥ 0.90.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find k:**<BR>\n" );
document.write( "    * We are given that |X̄ - μ| ≤ σ/2.<BR>\n" );
document.write( "    * Comparing this with |X̄ - μ| < kσ/√n, we have:<BR>\n" );
document.write( "        * kσ/√n = σ/2<BR>\n" );
document.write( "        * k/√n = 1/2<BR>\n" );
document.write( "        * k = √n / 2\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Use Chebyshev's Inequality:**<BR>\n" );
document.write( "    * 1 - 1/k² ≥ 0.90<BR>\n" );
document.write( "    * 1 - 0.90 ≥ 1/k²<BR>\n" );
document.write( "    * 0.10 ≥ 1/k²<BR>\n" );
document.write( "    * k² ≥ 1/0.10 = 10<BR>\n" );
document.write( "    * k ≥ √10\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Substitute k:**<BR>\n" );
document.write( "    * √n / 2 ≥ √10<BR>\n" );
document.write( "    * √n ≥ 2√10<BR>\n" );
document.write( "    * n ≥ (2√10)²<BR>\n" );
document.write( "    * n ≥ 4 * 10<BR>\n" );
document.write( "    * n ≥ 40\r<BR>\n" );
document.write( "\n" );
document.write( "**Conclusion**\r<BR>\n" );
document.write( "\n" );
document.write( "The sample size should be at least 40 so that we can be 90% certain that the sample mean X̄ will not deviate from the true mean μ by more than σ/2.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );