Answer to Question #137998 in Statistics and Probability for Manishi

Question

The standard deviation of lifetime of electric light bulbs is 100 hours. Find the 95% confidence limits for the population standard deviation for such electric bulbs.

Answer 1

Confidence intervals for the true standard deviation can be constructed using the chi-square distribution.

if the original population of data is normally distributed, then the expression $\dfrac{s^2(n-1)}{\sigma^2}$ has a chi-square distribution with  $n-1$ degrees of freedom.

The 100(1−α)% confidence intervals

Two-sided $(1-\alpha)\%$ confidence interval for $\sigma$ is

$$\dfrac{s\sqrt{n-1}}{\chi^2_{\alpha/2,n-1}}\leq \sigma\leq\dfrac{s\sqrt{n-1}}{\chi^2_{1-\alpha/2,n-1}}$$

The sample must be taken from a normally distributed population.

If $\alpha=0.05, n=50$

$\chi^2_{L}=\chi^2_{1-\alpha/2,n-1}=32.3574$

$\chi^2_{R}=\chi^2_{\alpha/2,n-1}=71.4202$

Given $s=100$

Two-sided $95\%$ confidence interval for $\sigma$ is

$$\dfrac{100\sqrt{50-1}}{71.4202}\leq \sigma\leq\dfrac{100\sqrt{50-1}}{32.3574}$$

$$9.8011\leq \sigma\leq21.6334$$