Test the claim about the population variance σ^2 or standard...

Test the claim about the population variance $\sigma^{2}$ or standard deviation $\sigma$ at the level of significance $\alpha .$ Assume the population is normally distributed.
Claim: $\sigma<40 ; \alpha=0.01 .$ Sample statistics: $s=40.8, n=12$

Key Concepts

Hypothesis Testing

This is the framework used to make inferences about a population parameter based on sample data. It involves setting up a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis (what the test aims to support), then using sample data to decide whether there is enough evidence to reject the null hypothesis. In testing variance or standard deviation, the hypotheses are framed in terms of the population variance, and the decision is based on a predetermined significance level.

Chi-Square Distribution

When a population is normally distributed, the test statistic used for variance (or standard deviation) testing follows a chi-square distribution. The statistic is computed as (n-1)s² divided by the hypothesized variance (??²), where n is the sample size and s² is the sample variance. This distribution is used to determine the critical value and to establish the rejection region for the test.

Variance Testing

Variance testing is a specific type of hypothesis test aimed at determining whether the variability in a population meets a given specification. In such tests, the sample variance is compared against a hypothesized variance. Depending on whether the claim is about the variance being greater than, less than, or not equal to a specific value, the rejection region is defined accordingly, often using a one-tailed or two-tailed test.

Level of Significance

The level of significance, denoted by alpha (?), is the probability threshold for rejecting the null hypothesis when it is true (a type I error). In variance testing, the choice of ? determines the critical value from the chi-square distribution that defines the decision rule or rejection region. A lower ?, such as 0.01, means that stronger evidence is required to reject the null hypothesis.

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