Question

( In a market study for BGI, a local department store, you select a sample of 60 actual and potential clients to interview. Among the questions you wish to answer is whether the clients and non-clients differ in their incomes. The table below gives summary statistics. Noting the rather large difference in sample standard deviations, you decide that you must assume that the population standard deviations are unequal. Can you conclude that there is a difference in the mean incomes of clients and non-clients? Use a = 0.05. Non-Clients Clients 58.7 16.8 Mean income in $1000s) Standard deviations (in $1000s) 50.4 9.8 Number 27 25

Answer 1

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.

Answer 2

To see if there's a significant difference between the mean incomes of clients and non-clients at a department store, a two-sample t-test for unequal variances is used with a 0.05 significance level. If the test statistic calculated is greater than the critical value, or if the p-value is less than 0.05, the null hypothesis that there is no difference in means is rejected, suggesting a significant difference.

Step-by-step explanation:

Given the non-clients have a mean income of $58.7K with a standard deviation of $16.8K, and clients have a mean income of $50.4K with a standard deviation of $9.8K, we can use a two-sample t-test to determine if there is a significant difference between the two means, assuming unequal variances (since the standard deviations are quite different). We will use a significance level of α = 0.05 for this hypothesis test.

First, the null hypothesis (H0) is that there is no difference in mean incomes between the groups (clients and non-clients), while the alternative hypothesis (Ha) is that there is a difference.

To perform the test, we calculate the test statistic using the formula for the two-sample t-test with unequal variances, then compare this test statistic to the critical t-value from the t-distribution table or use a p-value approach. If the calculated t is greater than the critical t or if the p-value is less than 0.05, we reject the null hypothesis, indicating a significant difference in mean incomes.