9.13:
Testing a Claim about Standard Deviation
Correct pricing of gold requires an accurate scale, and its accuracy is achieved by reducing the standard deviation of the mean weight.
Consider an example where a company claims that they have significantly reduced the standard deviation of their scales from 0.005 g to 0.003 g tested over 30 individual units.
To test this claim, a hypothesis test is conducted where the null hypothesis states that the old and improved models have an equal standard deviation. The alternative hypothesis states that the improved model is more accurate and has a significantly smaller standard deviation than the old model.
Testing the hypothesis requires the sample statistic to be converted to the Χ2 statistic as follows.
Here, the critical region at a 0.05 significance level falls at the left tail of the curve.
Observe that the Χ2 value calculated from the sample falls within it.
Also, the P-value obtained using the left-tailed test is less than 0.05.
So, the improved model proves significantly more accurate than the old model based on the test result.
9.13:
Testing a Claim about Standard Deviation
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.