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: $\sigma^{2}=0.52 ; \alpha=0.05 .$ Sample statistics: $s^{2}=0.508, n=18$

Key Concepts

Significance Level (?)

The significance level, denoted by ?, defines the threshold for making a decision in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, ? = 0.05 means there is a 5% risk of committing such an error, which is commonly used in statistical tests.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the claim being tested—in this case, that the population variance is equal to a specified value—while the alternative hypothesis (H1) represents a deviation from that claim. This framework allows researchers to use sample data to determine whether there is enough evidence to reject H0 in favor of H1.

Chi-Square Test for Variance

When the population is normally distributed, the chi-square test is a common method used to assess the claim about the population variance. The test statistic is calculated using the formula ?² = (n - 1)s²/??², where s² is the sample variance, n is the sample size, and ??² is the variance under H0. This statistic follows a chi-square distribution with n - 1 degrees of freedom if the null hypothesis is true.

Degrees of Freedom

Degrees of freedom in the context of testing the population variance are determined by the number of independent values that can vary in the calculation of the statistic, which is n - 1. This parameter is crucial in determining the shape of the chi-square distribution against which the test statistic is compared.

Test Statistic Calculation

Calculating the test statistic involves using the formula derived for the chi-square test. The computed value is then compared to the critical value from the chi-square distribution corresponding to n - 1 degrees of freedom and the chosen significance level. This comparison determines whether the null hypothesis is rejected or not.

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