Question

Normality Requirement What is different about the normality requirement for a confidence interval estimate of $\sigma$and the normality requirement for a confidence interval estimate of $\mu$?

Short answer

The normality requirement for estimating $\sigma$ is quite robust and cannot be moulded with large sample size, unlike the case for estimating$\mu$

Step-by-step solution

Given information

The confidence interval estimates of $\sigma$and $\mu$are considered.

Normality Requirement for estimating  σ

To construct a confidence interval estimate of $\sigma$, the population from which the sample is taken should be strictly normally distributed, even if the sample is sufficiently large.

Normality Requirement for estimating  μ

To construct a confidence interval estimate of $\mu$, the population from which the sample is taken should either be normally distributed or the sample size should be greater than 30.

Difference

The difference in the normality requirement for estimating $\sigma$ and $\mu$is that in case of estimating the population standard deviation is that the need of a normally distributed is quite rigorous and cannot be compromised. Also, in the case of estimating the population mean, if the sample is large, the method can be applied.

In simple words, if the population is not normally distributed and a sample of size greater than 30 is considered, then the confidence interval estimate of $\sigma$ cannot be constructed. Whereas, a confidence interval estimate of $\mu$ can be constructed.