Construct the indicated confidence intervals for (a) the...

Construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Assume the sample is from a normally distributed population.
$$c=0.95, s^{2}=11.56, n=30$$

Key Concepts

Normal Distribution Assumption

When constructing confidence intervals for parameters such as variance or standard deviation, it is often assumed that the data come from a normally distributed population. This assumption is critical because the distributions of the sample statistics (like the sample variance) are derived under the normality condition, specifically leading to the chi-square distribution for a scaled version of the sample variance.

Sample Variance

The sample variance is the statistic computed from sample data that estimates the population variance. It is a crucial measure in inferential statistics because it provides the basis for constructing confidence intervals for the actual variability in the population when combined with the chi-square distribution.

Chi-Square Distribution

The chi-square distribution plays a central role in constructing confidence intervals for the population variance of a normally distributed variable. When a sample of size n is drawn from a normal population, the statistic (n-1)(sample variance)/population variance follows a chi-square distribution with n-1 degrees of freedom, which is then used to derive the confidence interval.

Degrees of Freedom

Degrees of freedom, which typically equal n-1 for sample variance, refer to the number of independent values that can vary in the calculation of a statistic. In the context of the chi-square distribution used in confidence interval estimation for variance, the degrees of freedom determine the specific form of the chi-square distribution that is used for constructing the interval.

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