A complete procedure to test a claim about population standard deviation or population variance is explained here.

The hypothesis testing for the claim of population standard deviation (or variance) requires the data and samples to be random and unbiased. The population distribution also must be normal. There is no specific requirement on the sample size as the estimation is based on the chi-square distribution.

As a first step, the hypothesis (null and alternative) concerning the claim about population SD (or variance) should be stated clearly and expressed symbolically. The hypothesis generally claim a certain value of SD or variance to be tested. Samples provide sample SD or variance. Using both these values, the test statistic is calculated.

The critical value here depends on the sample size (or the degrees of freedom) calculated from the chi-square distribution. Based on the directionality in the alternative hypothesis, the test can be left-tailed, right-tailed, or two-tailed. The sample test statistic is compared with the critical chi-square value generally calculated at a 95% or 99% confidence level. Otherwise, P-value is obtained and compared with the significance level of 0.05 or 0.01 to conclude the hypothesis test.