document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179489>Question 1179489</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The height in centimeters of a certain plant is normally distributed. A random sample<BR>\n" );
document.write( "of the plant is measured and the results are as follows:\r<BR>\n" );
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document.write( "14.6;12.5;15.3; 16.1; 14.4; 12.9; 13.7; 14.9<BR>\n" );
document.write( "a. Find a point estimate for the mean height of this plant.<BR>\n" );
document.write( "b. Construct a 90% confidence interval for the true mean height of this particular<BR>\n" );
document.write( "plant. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850227 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=850227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Point Estimate for the Mean Height**\r<BR>\n" );
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document.write( "The sample mean (x̄) is a good point estimate for the population mean (μ).  \r<BR>\n" );
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document.write( "To calculate the sample mean, add up all the heights and divide by the number of plants in the sample:\r<BR>\n" );
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document.write( "x̄ = (14.6 + 12.5 + 15.3 + 16.1 + 14.4 + 12.9 + 13.7 + 14.9) / 8 <BR>\n" );
document.write( "x̄ = 114.4 / 8<BR>\n" );
document.write( "**x̄ = 14.3 cm**\r<BR>\n" );
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document.write( "**b) 90% Confidence Interval for the True Mean Height**\r<BR>\n" );
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document.write( "Since the population standard deviation is unknown, we'll use a t-distribution to construct the confidence interval.\r<BR>\n" );
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document.write( "**1. Calculate the Sample Standard Deviation (s)**\r<BR>\n" );
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document.write( "s = √[ Σ(xi - x̄)² / (n - 1) ] \r<BR>\n" );
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document.write( "where:<BR>\n" );
document.write( "  * xi = each individual height<BR>\n" );
document.write( "  * x̄ = sample mean <BR>\n" );
document.write( "  * n = sample size\r<BR>\n" );
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document.write( "s ≈ 1.283 cm  (You can use a calculator or software to compute this)\r<BR>\n" );
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document.write( "**2. Determine the Degrees of Freedom (df)**\r<BR>\n" );
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document.write( "df = n - 1 = 8 - 1 = 7\r<BR>\n" );
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document.write( "**3. Find the t-value**\r<BR>\n" );
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document.write( "For a 90% confidence interval and 7 degrees of freedom, we need the t-value that leaves 5% in each tail (α/2 = 0.10/2 = 0.05).\r<BR>\n" );
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document.write( "Using a t-table or calculator, the t-value is approximately 1.895.\r<BR>\n" );
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document.write( "**4. Calculate the Margin of Error (E)**\r<BR>\n" );
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document.write( "E = t * (s / √n) = 1.895 * (1.283 / √8) ≈ 0.861 cm\r<BR>\n" );
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document.write( "**5. Construct the Confidence Interval**\r<BR>\n" );
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document.write( "Confidence Interval = x̄ ± E = 14.3 ± 0.861\r<BR>\n" );
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document.write( "Lower Bound = 14.3 - 0.861 = 13.439 cm<BR>\n" );
document.write( "Upper Bound = 14.3 + 0.861 = 15.161 cm\r<BR>\n" );
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document.write( "**Therefore, the 90% confidence interval for the true mean height of the plant is approximately (13.44 cm, 15.16 cm).** </SPAN>\n" );
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