Final answer:
To determine if there is a difference in the mean incomes of clients and non-clients, a two-sample t-test for independent samples with unequal variances is conducted. The calculated t-statistic is compared with the critical t-value to determine if the null hypothesis should be rejected. In this case, there is a significant difference in the mean incomes of clients and non-clients.
Step-by-step explanation:
To determine if there is a difference in the mean incomes of clients and non-clients, we need to conduct a hypothesis test. Since the population standard deviations are assumed to be unequal, we will use a two-sample t-test for independent samples with unequal variances.
The null hypothesis (H0) states that there is no difference in the mean incomes between clients and non-clients. The alternative hypothesis (Ha) states that there is a difference.
We will calculate the t-statistic using the formula: t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2)), where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
With the given data, the calculated t-statistic is -2.417. We compare this with the critical t-value obtained from a t-table at a significance level of 0.05 and degrees of freedom equal to (n1 - 1) + (n2 - 1). If the calculated t-statistic is less than the critical t-value, we reject the null hypothesis.
In this case, the calculated t-statistic (-2.417) is greater than the critical t-value at a significance level of 0.05. Therefore, we can conclude that there is a significant difference in the mean incomes of clients and non-clients.