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Use technology to construct the confidence intervals for the population variance σ2\sigma^2 and the population standard deviation σ\sigma. Assume the sample is taken from a normally distributed population.\newlinec=0.99c=0.99, s=33s=33, n=16n=16\newlineThe confidence interval for the population variance is \newline□\square, \newline□\square. (Round to two decimal places as needed.)

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Q. Use technology to construct the confidence intervals for the population variance σ2\sigma^2 and the population standard deviation σ\sigma. Assume the sample is taken from a normally distributed population.\newlinec=0.99c=0.99, s=33s=33, n=16n=16\newlineThe confidence interval for the population variance is \newline□\square, \newline□\square. (Round to two decimal places as needed.)
  1. Calculate degrees of freedom: Use the chi-square distribution to find the critical values for the confidence interval of the population variance. The degrees of freedom (df) is n−1n - 1.df=n−1=16−1=15df = n - 1 = 16 - 1 = 15
  2. Find chi-square critical values: Find the chi-square critical values using a chi-square table or technology for df=15df = 15 and c=0.99c = 0.99. The lower critical value (chi-square left) corresponds to the upper tail and the upper critical value (chi-square right) corresponds to the lower tail because the chi-square distribution is not symmetric.\newlineχ2(0.005,15)\chi^2(0.005, 15)\newlineχ2(0.995,15)\chi^2(0.995, 15)
  3. Use formula for confidence interval: Plug the critical values and the sample standard deviation into the formula for the confidence interval of the population variance.\newlineConfidence interval for variance = [(n−1)⋅s2]/χright2,[(n−1)⋅s2]/χleft2[\left(n - 1\right) \cdot s^2] / \chi^2_{\text{right}}, [\left(n - 1\right) \cdot s^2] / \chi^2_{\text{left}}
  4. Calculate confidence interval: Calculate the confidence interval using the sample standard deviation s=33s = 33 and the sample size n=16n = 16. Confidence interval for variance = [15×(33)2]/χright2,[15×(33)2]/χleft2[15 \times (33)^2] / \chi^2_{\text{right}}, [15 \times (33)^2] / \chi^2_{\text{left}}
  5. Find exact chi-square critical values: Use technology to find the exact chi-square critical values.\newlineχ2(0.005,15)≈5.23\chi^2(0.005, 15) \approx 5.23\newlineχ2(0.995,15)≈30.58\chi^2(0.995, 15) \approx 30.58
  6. Substitute critical values: Substitute the critical values into the confidence interval formula.\newlineConfidence interval for variance = [(15)×(33)2]/30.58[\left(15\right) \times \left(33\right)^{2}] / 30.58, [(15)×(33)2]/5.23[\left(15\right) \times \left(33\right)^{2}] / 5.23
  7. Perform calculations: Perform the calculations to find the confidence interval for the population variance.\newlineConfidence interval for variance = [(15)×(1089)30.58,(15)×(1089)5.23][\frac{(15) \times (1089)}{30.58}, \frac{(15) \times (1089)}{5.23}]\newlineConfidence interval for variance = [1633530.58,163355.23][\frac{16335}{30.58}, \frac{16335}{5.23}]\newlineConfidence interval for variance ≈534.16,3124.28\approx 534.16, 3124.28
  8. Round variance interval: Round the confidence interval for the population variance to two decimal places.\newlineConfidence interval for variance ≈534.16,3124.28\approx 534.16, 3124.28
  9. Calculate square roots: To find the confidence interval for the population standard deviation, take the square root of the variance interval endpoints.\newlineConfidence interval for standard deviation = 534.16\sqrt{534.16}, 3124.28\sqrt{3124.28}
  10. Round standard deviation interval: Calculate the square roots.\newlineConfidence interval for standard deviation ≈23.11,55.89\approx 23.11, 55.89
  11. Round standard deviation interval: Calculate the square roots.\newlineConfidence interval for standard deviation ≈23.11,55.89\approx 23.11, 55.89Round the confidence interval for the population standard deviation to two decimal places.\newlineConfidence interval for standard deviation ≈23.11,55.89\approx 23.11, 55.89

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