document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178025>Question 1178025</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following observation in a simple random sample from a population that is approximately normally distributed. Construct and interpret a 90% confidence interval for the mean 67,79,71,98,74,70,59,102, 92,96 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850377 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=850377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's construct and interpret a 90% confidence interval for the mean of the given sample.\r<BR>\n" );
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document.write( "**1. Calculate the Sample Mean (x̄) and Sample Standard Deviation (s)**\r<BR>\n" );
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document.write( "* **Data:** 67, 79, 71, 98, 74, 70, 59, 102, 92, 96<BR>\n" );
document.write( "* **Sample Size (n):** 10\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy import stats\r<BR>\n" );
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document.write( "data = [67, 79, 71, 98, 74, 70, 59, 102, 92, 96]<BR>\n" );
document.write( "x_bar = np.mean(data)<BR>\n" );
document.write( "s = np.std(data, ddof=1)  # ddof=1 for sample standard deviation\r<BR>\n" );
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document.write( "print(f\"Sample Mean (x̄): {x_bar:.2f}\")<BR>\n" );
document.write( "print(f\"Sample Standard Deviation (s): {s:.2f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Output:\r<BR>\n" );
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document.write( "* Sample Mean (x̄): 80.80<BR>\n" );
document.write( "* Sample Standard Deviation (s): 14.67\r<BR>\n" );
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document.write( "**2. Degrees of Freedom**\r<BR>\n" );
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document.write( "* df = n - 1 = 10 - 1 = 9\r<BR>\n" );
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document.write( "**3. Find the Critical Value (t*)**\r<BR>\n" );
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document.write( "* **Confidence Level:** 90%<BR>\n" );
document.write( "* **Alpha (α):** 1 - 0.90 = 0.10<BR>\n" );
document.write( "* **Alpha/2:** α/2 = 0.05<BR>\n" );
document.write( "* **Critical Value (t*):** Using a t-distribution table or calculator with df = 9 and α/2 = 0.05, we find t* ≈ 1.833.\r<BR>\n" );
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document.write( "**4. Calculate the Standard Error (SE)**\r<BR>\n" );
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document.write( "* **Standard Error (SE):** SE = s / √n<BR>\n" );
document.write( "* SE = 14.67 / √10 ≈ 14.67 / 3.1623 ≈ 4.639\r<BR>\n" );
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document.write( "**5. Calculate the Margin of Error (ME)**\r<BR>\n" );
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document.write( "* **Margin of Error (ME):** ME = t* * SE<BR>\n" );
document.write( "* ME = 1.833 * 4.639 ≈ 8.504\r<BR>\n" );
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document.write( "**6. Construct the Confidence Interval**\r<BR>\n" );
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document.write( "* **Confidence Interval:** x̄ ± ME<BR>\n" );
document.write( "* **Lower Bound:** x̄ - ME = 80.80 - 8.504 ≈ 72.296<BR>\n" );
document.write( "* **Upper Bound:** x̄ + ME = 80.80 + 8.504 ≈ 89.304\r<BR>\n" );
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document.write( "**Therefore, the 90% confidence interval is (72.30, 89.30).**\r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "We are 90% confident that the true population mean lies within the interval (72.30, 89.30). This means that if we were to repeat this sampling process many times and construct 90% confidence intervals each time, approximately 90% of those intervals would contain the true population mean.<BR>\n" );
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