document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178049>Question 1178049</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Twelve different video games showing substance use were observed and the duration of times of gameplay (in seconds) are listed below. The design of the study justifies the assumption that the sample can be treated as a simple random sample. Use the sample data to construct a 90% <BR>\n" );
document.write( "confidence interval estimate of σ, the standard deviation of the duration times of gameplay. Assume that this sample was obtained from a population with a normal distribution.<BR>\n" );
document.write( "4,665	4,432	4,811	4,393	4,242    4,969	 <BR>\n" );
document.write( "4,100	4,721	4,864	4,244	4,364	4,254	\r<BR>\n" );
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document.write( "The confidence interval estimate is <BR>\n" );
document.write( "____ sec<σ<__sec.<BR>\n" );
document.write( "(Round to one decimal place as needed.)\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850372 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=850372\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\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( "First, we need to calculate the sample mean (x̄) and the sample standard deviation (s).\r<BR>\n" );
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document.write( "* **Data:** 4665, 4432, 4811, 4393, 4242, 4969, 4100, 4721, 4864, 4244, 4364, 4254<BR>\n" );
document.write( "* **Sample Size (n):** 12\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "data = [4665, 4432, 4811, 4393, 4242, 4969, 4100, 4721, 4864, 4244, 4364, 4254]<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): 4496.67<BR>\n" );
document.write( "* Sample Standard Deviation (s): 291.60\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 = 12 - 1 = 11\r<BR>\n" );
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document.write( "**3. Chi-Square Values**\r<BR>\n" );
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document.write( "We need to find the chi-square values for the lower and upper bounds of the confidence interval.\r<BR>\n" );
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document.write( "* **Confidence Level:** 90% (0.90)<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( "* **1 - Alpha/2:** 1 - 0.05 = 0.95\r<BR>\n" );
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document.write( "We'll use the chi-square distribution with 11 degrees of freedom.\r<BR>\n" );
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document.write( "* **χ²_lower:** χ²(0.95, 11)<BR>\n" );
document.write( "* **χ²_upper:** χ²(0.05, 11)\r<BR>\n" );
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document.write( "Using a chi-square table or calculator:\r<BR>\n" );
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document.write( "* χ²_lower ≈ 4.575<BR>\n" );
document.write( "* χ²_upper ≈ 19.675\r<BR>\n" );
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document.write( "**4. Calculate the Confidence Interval for σ**\r<BR>\n" );
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document.write( "The confidence interval for the population standard deviation (σ) is given by:\r<BR>\n" );
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document.write( "* √[(n - 1) * s² / χ²_upper] < σ < √[(n - 1) * s² / χ²_lower]\r<BR>\n" );
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document.write( "Let's plug in the values:\r<BR>\n" );
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document.write( "* √[(11 * 291.60²) / 19.675] < σ < √[(11 * 291.60²) / 4.575]<BR>\n" );
document.write( "* √[(11 * 85030.56) / 19.675] < σ < √[(11 * 85030.56) / 4.575]<BR>\n" );
document.write( "* √[935336.16 / 19.675] < σ < √[935336.16 / 4.575]<BR>\n" );
document.write( "* √47534.24 < σ < √204444.95<BR>\n" );
document.write( "* 218.0 < σ < 452.1\r<BR>\n" );
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document.write( "Rounded to one decimal place:\r<BR>\n" );
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document.write( "* 218.0 sec < σ < 452.1 sec\r<BR>\n" );
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document.write( "**Therefore, the 90% confidence interval estimate of σ is 218.0 sec < σ < 452.1 sec.**<BR>\n" );
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