Calculate the value for the test statistic, χ^2 ⋆, for each of these situations: a. Ho: σ^2=20,

step 1 Hi, to find the statistics here, you have to follow this formula. In here is the sample size. Th

Calculate the value for the test statistic, χ^2 ⋆, for each...

Calculate the value for the test statistic, $\chi^{2} \star,$ for each of these situations:
a. $H_{o}: \sigma^{2}=20, n=15, s^{2}=17.8$
b. $H_{o}: \sigma^{2}=30, n=18, s=5.7$
c. $H_{o}: \sigma=42, n=25, s=37.8$
d. $H_{o}: \sigma=12, n=37, s^{2}=163$

Key Concepts

Hypothesis Testing for Variance

Hypothesis testing for variance involves formulating a null hypothesis that specifies the population variance (or standard deviation) and comparing it with the observed sample variance. By computing a test statistic using the chi-square formula, one can assess whether deviations from the hypothesized value are statistically significant. This process is crucial in determining if the sample provides enough evidence to reject the null hypothesis about the population variance.

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution that arises often in statistics, particularly in tests of variance. It is used to determine the distribution of a sum of squared independent standard normal random variables, and its shape depends on the degrees of freedom associated with the test. In hypothesis testing, the chi-square distribution is fundamental when evaluating claims about the variability of a normally distributed population.

Test Statistic for Variance

The test statistic for variance in a normally distributed population is calculated as (n - 1)s² divided by the hypothesized variance ??². This statistic follows a chi-square distribution with n - 1 degrees of freedom under the null hypothesis. It provides a standardized way to compare the sample variance with the population variance claimed under the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available to estimate another parameter. In the context of the chi-square test for variance, the degrees of freedom are equal to n - 1, where n is the sample size. This adjustment accounts for the estimation of the sample variance, which consumes one degree of freedom.

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