document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174397>Question 1174397</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sample of 60 grade 9 students' ages was obtained to estimate the mean age of all grade 9 students'.sample mean is equal to 15.3 years and the population variance is 16. What is the point estimate for mean and Find the 95% confidence interval for mean. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850671 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Point Estimate for the Mean:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The sample mean (x̄) is the best point estimate for the population mean (μ).<BR>\n" );
document.write( "* Therefore, the point estimate for the mean age of all grade 9 students is **15.3 years**.\r<BR>\n" );
document.write( "\n" );
document.write( "**95% Confidence Interval for the Mean:**\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Determine the Critical Value:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Since the population variance is known and the sample size is relatively large (n > 30), we can use the z-distribution.<BR>\n" );
document.write( "* For a 95% confidence interval, the alpha level (α) is 1 - 0.95 = 0.05.<BR>\n" );
document.write( "* Since we're constructing a two-tailed confidence interval, we divide alpha by 2: α/2 = 0.025.<BR>\n" );
document.write( "* The z-score corresponding to a cumulative probability of 0.975 (1 - 0.025) in a standard normal distribution table or using a calculator is approximately 1.96.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Calculate the Margin of Error:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The margin of error (E) is calculated as:<BR>\n" );
document.write( "    E = z * (σ / √n)<BR>\n" );
document.write( "    where:<BR>\n" );
document.write( "      * z is the critical value (1.96)<BR>\n" );
document.write( "      * σ is the population standard deviation (√population variance = √16 = 4)<BR>\n" );
document.write( "      * n is the sample size (60)\r<BR>\n" );
document.write( "\n" );
document.write( "* E = 1.96 * (4 / √60) ≈ 1.01\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Construct the Confidence Interval:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The confidence interval is calculated as:<BR>\n" );
document.write( "   (x̄ - E, x̄ + E)\r<BR>\n" );
document.write( "\n" );
document.write( "* (15.3 - 1.01, 15.3 + 1.01)<BR>\n" );
document.write( "* (14.29, 16.31)\r<BR>\n" );
document.write( "\n" );
document.write( "**Interpretation:**\r<BR>\n" );
document.write( "\n" );
document.write( "We are 95% confident that the true mean age of all grade 9 students falls between 14.29 years and 16.31 years. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );