Question

A nationwide job recruiting firm wants to compare the annual incomes for childcare workers in California and Oregon. Due to recent trends in the childcare industry, the firm suspects that the that the mean annual income of childcare workers in the state of California is greater than the mean annual income of childcare workers in Oregon. To see if this is true, the firm selected a random sample of 10 childcare workers from California and an independent random sample of to childcare workers from Oregon and asked them to report their mean annual income. The data obtained were as follows. The population standard deviations for the annual incomes of childcare workers in California and in Oregon are estimated as 6100 and 6200, respectively. It is also known that both populations are approximately normally distributed. At the 0.05 level of significance, is there sufficient evidence to support the claim that the mean annual income, μ1, of childcare workers in California is greater than the mean annual income, μ2, of childcare workers in Oregon? Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to at least three decimal places. (If necessary, consult a list of formulas.) (a) State the noll hupethesis H0 and the asfernative hroothesis H1 H0=H1 nd the p-value. (Round to three or more decimal places.) (e) Can we support the caaim that the mean annual income of childcare workers in California is greater than the mean annual income of childcare workers in Oregen? Yes No

Answer 1

To determine if the mean annual income of childcare workers in California is greater than in Oregon, we state the null and alternative hypotheses, calculate a test statistic, and make a decision based on the p-value. If the p-value is less than 0.05, we reject the null and conclude that there is sufficient evidence of a difference in mean incomes.

Step-by-step explanation:

The task here is to conduct a hypothesis test to determine if there is a significant difference in the mean annual incomes of childcare workers in California (μ1) and Oregon (μ2). With an alpha of 0.05, we are conducting a one-tailed test because the suspicion is that the mean annual income in California is greater than in Oregon.

Step A - Hypotheses

H0 (null hypothesis): μ1 ≤ μ2
HA (alternative hypothesis): μ1 > μ2

Step B - Test Statistic and Decision

To test the hypothesis, we would calculate the test statistic using the given sample means, sample sizes, and population standard deviations. Since the population standard deviations are known, we would use a Z-test for the test statistic. If the p-value calculated from the Z-test statistic is less than the significance level of 0.05, then we reject the null hypothesis.

Step C - Conclusion

Based on the decision to reject the null hypothesis (assuming the calculated p-value is < 0.05), there is sufficient evidence to support the claim that the mean annual income of childcare workers in California is greater than that of childcare workers in Oregon. This conclusion mirrors the provided example that at the 5 percent significance level, there is sufficient evidence to conclude that a mean salary exceeds a given number.