Math Problem Statement

An optical firm purchases glass for making lenses. Assume that the refractive index of 20 pieces

of glass have a variance of 1.2*10^-4

. Construct a 95% confidence interval for , the standard

deviation of the population samples.

Solution

To construct a 95% confidence interval for the standard deviation of the population, we will use the Chi-square distribution. The formula to calculate the confidence interval for the population standard deviation σ\sigma is:

((n1)s2χα/22,(n1)s2χ1α/22)\left( \sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{1-\alpha/2}}} \right)

Where:

  • nn is the sample size (in this case, n=20n = 20),
  • s2s^2 is the sample variance (given as 1.2×1041.2 \times 10^{-4}),
  • α\alpha is the significance level (for a 95% confidence interval, α=0.05\alpha = 0.05),
  • χα/22\chi^2_{\alpha/2} and χ1α/22\chi^2_{1-\alpha/2} are the critical values of the chi-square distribution with n1n-1 degrees of freedom.

Steps

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Math Problem Analysis

Mathematical Concepts

Statistics
Chi-square distribution
Confidence intervals
Variance and standard deviation

Formulas

Chi-square confidence interval for standard deviation formula: (sqrt((n-1) * s^2 / chi^2_alpha/2), sqrt((n-1) * s^2 / chi^2_1-alpha/2))

Theorems

Chi-square distribution theorem for variance and standard deviation estimation

Suitable Grade Level

Grades 11-12