document.write( "Question 1168610: Listed below are speeds​ (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at​ 3:30 P.M. on a weekday. Use the sample data to construct 95​% confidence interval estimate of the population standard deviation.
\n" ); document.write( "62,63,63,56,63,52,61,60,61,69,60,66
\n" ); document.write( "Need to find
\n" ); document.write( "The confidence interval estimate is mi/h < o < mi/h
\n" ); document.write( "(round to one decimal place as needed) \n" ); document.write( "
Algebra.Com's Answer #851515 by CPhill(1959)\"\" \"About 
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Let's calculate the 95% confidence interval for the population standard deviation.\r
\n" ); document.write( "\n" ); document.write( "**1. Calculate the Sample Standard Deviation (s)**\r
\n" ); document.write( "\n" ); document.write( "First, we need to find the sample mean and then the sample standard deviation.\r
\n" ); document.write( "\n" ); document.write( "* Data: 62, 63, 63, 56, 63, 52, 61, 60, 61, 69, 60, 66
\n" ); document.write( "* n = 12\r
\n" ); document.write( "\n" ); document.write( "* Sample mean (x̄):
\n" ); document.write( " * Sum = 62 + 63 + 63 + 56 + 63 + 52 + 61 + 60 + 61 + 69 + 60 + 66 = 736
\n" ); document.write( " * x̄ = 736 / 12 ≈ 61.33\r
\n" ); document.write( "\n" ); document.write( "* Sample standard deviation (s):
\n" ); document.write( " * s = √[Σ(xᵢ - x̄)² / (n - 1)]
\n" ); document.write( " * Calculating the differences:
\n" ); document.write( " * (62 - 61.33)² ≈ 0.4489
\n" ); document.write( " * (63 - 61.33)² ≈ 2.7889
\n" ); document.write( " * (63 - 61.33)² ≈ 2.7889
\n" ); document.write( " * (56 - 61.33)² ≈ 28.4089
\n" ); document.write( " * (63 - 61.33)² ≈ 2.7889
\n" ); document.write( " * (52 - 61.33)² ≈ 86.9489
\n" ); document.write( " * (61 - 61.33)² ≈ 0.1089
\n" ); document.write( " * (60 - 61.33)² ≈ 1.7689
\n" ); document.write( " * (61 - 61.33)² ≈ 0.1089
\n" ); document.write( " * (69 - 61.33)² ≈ 58.8289
\n" ); document.write( " * (60 - 61.33)² ≈ 1.7689
\n" ); document.write( " * (66 - 61.33)² ≈ 21.8089
\n" ); document.write( " * Sum of squares ≈ 208.5732
\n" ); document.write( " * s = √(208.5732 / 11) ≈ √18.9612 ≈ 4.354\r
\n" ); document.write( "\n" ); document.write( "**2. Determine the Chi-Square Values**\r
\n" ); document.write( "\n" ); document.write( "* Degrees of freedom (df) = n - 1 = 12 - 1 = 11
\n" ); document.write( "* Confidence level = 95%, so α = 0.05
\n" ); document.write( "* α/2 = 0.025
\n" ); document.write( "* 1 - α/2 = 0.975\r
\n" ); document.write( "\n" ); document.write( "* Using a chi-square distribution table or calculator:
\n" ); document.write( " * χ²(0.025, 11) ≈ 21.920
\n" ); document.write( " * χ²(0.975, 11) ≈ 3.816\r
\n" ); document.write( "\n" ); document.write( "**3. Calculate the Confidence Interval**\r
\n" ); document.write( "\n" ); document.write( "* Confidence interval for σ:
\n" ); document.write( " * √[(n - 1)s² / χ²(α/2)] < σ < √[(n - 1)s² / χ²(1 - α/2)]\r
\n" ); document.write( "\n" ); document.write( "* Lower bound:
\n" ); document.write( " * √[(11 * 4.354²) / 21.920] ≈ √[(11 * 18.9573) / 21.920] ≈ √(208.5303 / 21.920) ≈ √9.5132 ≈ 3.084\r
\n" ); document.write( "\n" ); document.write( "* Upper bound:
\n" ); document.write( " * √[(11 * 4.354²) / 3.816] ≈ √[(11 * 18.9573) / 3.816] ≈ √(208.5303 / 3.816) ≈ √54.6463 ≈ 7.392\r
\n" ); document.write( "\n" ); document.write( "**4. Round to One Decimal Place**\r
\n" ); document.write( "\n" ); document.write( "* Lower bound: 3.1
\n" ); document.write( "* Upper bound: 7.4\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the 95% confidence interval estimate is 3.1 mi/h < σ < 7.4 mi/h.**
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