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, $$ \sum_{i=1}^{9} x_{i}=333 \text { and } \sum_{i=1}^{9}\left(x_{i}-\bar{x}\right)^{2}=312 $$ where $x_{i}$ is measured in thousands of dollars and the population distribution can be assumed to be normal. Test, at the $5 \%$ level, the null hypothesis that the population mean is at least $$$ 40,000.$$

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,
$$
\sum_{i=1}^{9} x_{i}=333 \text { and } \sum_{i=1}^{9}\left(x_{i}-\bar{x}\right)^{2}=312
$$
where $x_{i}$ is measured in thousands of dollars and the population distribution can be assumed to be normal. Test, at the $5 \%$ level, the null hypothesis that the population mean is at least $$$ 40,000.$$

Key Concepts

Null and Alternative Hypotheses

The null hypothesis is a statement about the population parameter that the test is designed to challenge, while the alternative hypothesis represents the condition that is accepted if the null is rejected. In testing claims about a population mean, the null might assert that the mean is at least a specific value, whereas the alternative would state that it is less than that value, setting the stage for a one-tailed test.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating a null hypothesis (typically representing the status quo or a claim of no change) and an alternative hypothesis (representing the claim to be tested), then using a test statistic to determine whether the sample data provide sufficient evidence to reject the null hypothesis at a predetermined level of significance.

One-Sample t-Test

When the population variance is unknown and the sample size is small, the one-sample t-test is used to evaluate the hypothesis about the population mean. This test uses the sample standard deviation to estimate the standard error of the mean and compares the calculated t-statistic to a critical value from the t-distribution based on the appropriate degrees of freedom.

Test Statistic

The test statistic is a standardized value calculated from the sample data that measures the degree to which the sample mean deviates from the hypothesized population mean. For a one-sample t-test, the statistic is obtained by subtracting the hypothesized mean from the sample mean and then dividing by the standard error of the mean, which accounts for sample variability and size.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold used to determine whether the observed data is unlikely under the null hypothesis. A significance level of 5% means there is a 5% risk of rejecting the null hypothesis when it is actually true, which helps control for Type I errors in hypothesis testing.

Transcript

Please give Ace some feedback