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: $\sigma=24.9 ; \alpha=0.10 .$ Sample statistics: $s=29.1, n=51$

Key Concepts

Hypothesis Testing

Hypothesis testing involves evaluating a claim about a population parameter using sample data. In this context, the claim about the population standard deviation or variance is tested by comparing the sample statistic to what is expected under the null hypothesis.

Chi-square Test for Variance

The chi-square test for variance is a statistical method used when testing hypotheses about the variance (or standard deviation) of a normally distributed population. The test statistic is calculated as (n-1) times the sample variance divided by the hypothesized variance, and it follows a chi-square distribution.

Degrees of Freedom

Degrees of freedom, typically n-1 in variance tests, refer to the number of independent values that can vary in the calculation of a statistic. In the chi-square test for variance, they determine the shape of the chi-square distribution and are crucial for determining critical values.

Significance Level

The significance level, denoted by alpha, is a threshold used to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, that is, rejecting a true null hypothesis. In this problem, the level of significance is set to 0.10.

Normal Distribution Assumption

The chi-square test for variance assumes that the population from which the sample is drawn is normally distributed. This assumption is critical because the sampling distribution of the test statistic is derived under the normality condition.

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