An insurance company employs agents on a commission basis....

An insurance company employs agents on a commission basis. It claims that, in their first year, agents will earn a mean commission of at least $$\$ 40,000$$ and that the population standard deviation is no more than $$\$ 6,000 .$$ A random sample of nine agents found for commission in the first year, $$ \sum_{i=1}^{9} x_{i}=333,000 \quad \text { and } \quad \sum_{i=1}^{9}\left(x_{i}-\bar{x}\right)^{2}=312,000,000 $$ measured in thousands of dollars. The population distribution can be assumed to be normal. Test, at the $10 \%$ level, the null hypothesis that the population standard deviation is at most $$\$ 6,000.$$

An insurance company employs agents on a commission basis. It claims that, in their first year, agents will earn a mean commission of at least $$\$ 40,000$$ and that the population standard deviation is no more than $$\$ 6,000 .$$ A random sample of nine agents found for commission in the first year,
$$
\sum_{i=1}^{9} x_{i}=333,000 \quad \text { and } \quad \sum_{i=1}^{9}\left(x_{i}-\bar{x}\right)^{2}=312,000,000
$$
measured in thousands of dollars. The population distribution can be assumed to be normal. Test, at the $10 \%$ level, the null hypothesis that the population standard deviation is at most $$\$ 6,000.$$

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available for estimating a parameter. In the context of the chi-square test for variance, the degrees of freedom are determined by the sample size minus one (n-1), accounting for the estimation of the sample mean. This value is critical in identifying the correct chi-square distribution that determines the critical values used in the hypothesis test.

Significance Level and Critical Region

The significance level (commonly denoted as alpha, such as 10% in this example) is the threshold used to determine whether the observed test statistic is extreme enough to reject the null hypothesis. It defines the probability of making a Type I error. In a one-tailed test for variance, the critical region is located in the upper tail of the chi-square distribution. If the test statistic exceeds the chi-square critical value corresponding to the specified significance level, the null hypothesis is rejected.

Chi-Square Test for Variance

When samples are drawn from a normally distributed population, the variability of the data can be assessed using the chi-square test. In particular, the test statistic (n-1)*s²/??² follows a chi-square distribution with n-1 degrees of freedom, assuming the null hypothesis is true. This test is specifically designed to evaluate claims about the population variance or standard deviation.

Null and Alternative Hypotheses

In any hypothesis test, it is essential to clearly define the null hypothesis (H?) and the alternative hypothesis (H?). For tests concerning the amount of variation in a population, the null hypothesis might state that the population standard deviation or variance does not exceed a particular value, while the alternative hypothesis asserts that it is greater than that value. These hypotheses set the framework for the test and determine what constitutes evidence against the null hypothesis.

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