Which of the following is not true when testing a claim about a standard deviation or variance (${}^2$)?

• A. The population must be approximately normally distributed.
• B. The test statistic is based on the ${}^2$ (chi-square) distribution.
• C. Sample variance ($s^2$) is used to estimate the population variance (${}^2$).
• D. The null hypothesis always states that the variance (${}^2$) is less than a specific value.

Answer: (D)

Verified step by step guidance

1. Understand the context: When testing claims about a population variance or standard deviation, certain assumptions and procedures are standard.
2. Recall the assumptions: One key assumption is that the population from which the sample is drawn must be approximately normally distributed. This is important because the chi-square test for variance relies on this assumption.
3. Identify the test statistic: The test statistic used in hypothesis testing about variance or standard deviation is based on the chi-square distribution, which is derived from the sample variance and the population variance under the null hypothesis.
4. Recognize the role of sample variance: The sample variance is used as an estimator for the population variance in the test statistic calculation.
5. Analyze the null hypothesis statement: The null hypothesis in variance testing typically states that the population variance is equal to a specific value (not less than). It can be equal to, greater than, or less than that value depending on the claim, but it does not always state 'less than'. Therefore, the statement 'The null hypothesis always states that the variance is less than a specific value' is not true.