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.
62,63,63,56,63,52,61,60,61,69,60,66
Need to find
The confidence interval estimate is mi/h < o < mi/h
(round to one decimal place as needed)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's calculate the 95% confidence interval for the population standard deviation.
**1. Calculate the Sample Standard Deviation (s)**
First, we need to find the sample mean and then the sample standard deviation.
* Data: 62, 63, 63, 56, 63, 52, 61, 60, 61, 69, 60, 66
* n = 12
* Sample mean (x̄):
* Sum = 62 + 63 + 63 + 56 + 63 + 52 + 61 + 60 + 61 + 69 + 60 + 66 = 736
* x̄ = 736 / 12 ≈ 61.33
* Sample standard deviation (s):
* s = √[Σ(xᵢ - x̄)² / (n - 1)]
* Calculating the differences:
* (62 - 61.33)² ≈ 0.4489
* (63 - 61.33)² ≈ 2.7889
* (63 - 61.33)² ≈ 2.7889
* (56 - 61.33)² ≈ 28.4089
* (63 - 61.33)² ≈ 2.7889
* (52 - 61.33)² ≈ 86.9489
* (61 - 61.33)² ≈ 0.1089
* (60 - 61.33)² ≈ 1.7689
* (61 - 61.33)² ≈ 0.1089
* (69 - 61.33)² ≈ 58.8289
* (60 - 61.33)² ≈ 1.7689
* (66 - 61.33)² ≈ 21.8089
* Sum of squares ≈ 208.5732
* s = √(208.5732 / 11) ≈ √18.9612 ≈ 4.354
**2. Determine the Chi-Square Values**
* Degrees of freedom (df) = n - 1 = 12 - 1 = 11
* Confidence level = 95%, so α = 0.05
* α/2 = 0.025
* 1 - α/2 = 0.975
* Using a chi-square distribution table or calculator:
* χ²(0.025, 11) ≈ 21.920
* χ²(0.975, 11) ≈ 3.816
**3. Calculate the Confidence Interval**
* Confidence interval for σ:
* √[(n - 1)s² / χ²(α/2)] < σ < √[(n - 1)s² / χ²(1 - α/2)]
* Lower bound:
* √[(11 * 4.354²) / 21.920] ≈ √[(11 * 18.9573) / 21.920] ≈ √(208.5303 / 21.920) ≈ √9.5132 ≈ 3.084
* Upper bound:
* √[(11 * 4.354²) / 3.816] ≈ √[(11 * 18.9573) / 3.816] ≈ √(208.5303 / 3.816) ≈ √54.6463 ≈ 7.392
**4. Round to One Decimal Place**
* Lower bound: 3.1
* Upper bound: 7.4
**Therefore, the 95% confidence interval estimate is 3.1 mi/h < σ < 7.4 mi/h.**
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