Introduction
Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population variance or population standard deviation.
In this article we will discuss step by step procedure to construct a confidence interval for population variance or population standard deviation.
Confidence Interval for Variance
Let $X_1, X_2, \cdots , X_n$ be a random sample of size $n$ from $N(\mu, \sigma^2)$.
Let $\overline{X}=\frac{1}{n} \sum X_i$ be the sample mean and $s^2=\dfrac{1}{n-1}\bigg(\sum_{i=1}^nx_i^2-\dfrac{\big(\sum x_i\big)^2}{n}\bigg)$ be the sample variance.
Let $C=1-\alpha$ be the confidence coefficient.
$100(1-\alpha)$% confidence interval estimate of population variance $\sigma^2$ is
$$ \begin{aligned} \bigg(\frac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}, \frac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}\bigg) \end{aligned} $$
$100(1-\alpha)$% confidence interval estimate of population standard deviation $\sigma^2$ is
$$ \begin{aligned} \sqrt{\bigg(\frac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}}\bigg) \end{aligned} $$
Assumptions
a. The sample is a simple random sample.
b. The population has a normal distribution.
Step by step procedure
Step by step procedure to estimate confidence interval for population variance $\sigma^2$ is as follows:
Step 1 Specify the confidence level $(1-\alpha)$
Step 2 Given information
Specify the given information, sample size $n$, sample mean $\overline{X}$ and sample variance $s^2$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval estimate of population variance $\sigma^2$ is
$$ \begin{aligned} \bigg(\frac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}, \frac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}\bigg) \end{aligned} $$
Step 4 Determine the critical value
Determine the critical values $\chi^2_L = \chi^2_{(\alpha/2,n-1)}$ and $\chi^2_R = \chi^2_{(1-\alpha/2,n-1)}$ from $\chi^2$ statistical table that corresponds to the desired confidence level and the degrees of freedom.
Step 5 Determine the confidence interval
$100(1-\alpha)$% confidence interval estimate for population variance $\sigma^2$ is
$$ \begin{aligned} \bigg(\frac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}, \frac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}\bigg) \end{aligned} $$
