An auto manufacturer claims that the variance of the gas...

An auto manufacturer claims that the variance of the gas mileages in a certain vehicle model is $1.0 .$ A random sample of 25 vehicles has a variance of $1.65 .$ At $\alpha=0.05,$ is there enough evidence to reject the manufacturer's claim? (Adapted from Green Hybrid)

Key Concepts

Significance Level (?)

The significance level, often denoted by ?, is the probability threshold for rejecting the null hypothesis. It defines how extreme the test statistic must be, relative to the null hypothesis, to conclude that there is evidence against it. Commonly, a significance level of 0.05 is used, meaning that there is a 5% risk of incorrectly rejecting the null hypothesis when it is actually true.

Hypothesis Testing

In hypothesis testing, a claim about a population parameter is tested using sample data. The null hypothesis typically represents a statement of no difference or no effect (for example, that the population variance is equal to a claimed value), while the alternative hypothesis represents a deviation from that claim. The decision to reject or fail to reject the null hypothesis is based on the likelihood of observing the sample data if the null hypothesis were true.

Chi-Square Test for Variance

The chi-square test is used to assess whether the observed variance in a sample deviates significantly from a specified population variance. The test statistic is calculated by multiplying the degrees of freedom (sample size minus one) by the sample variance, then dividing by the hypothesized population variance. This statistic follows a chi-square distribution, which is used to determine the critical value for the test.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in the analysis after certain constraints are imposed (typically, the sample size minus one for variance tests). In the context of the chi-square test for variance, the degrees of freedom determine the exact shape of the chi-square distribution, which in turn affects the critical values used in the decision-making process.

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