What is the formula to do this exercise, Please.

A sample of 40 investment customers serviced by an account manager are found to have had an average of $23,000 in transactions during the past year, with a standard deviation of $8500. A sample of 30 customers serviced by another account manger averaged $28,000 in transaction, with a standard deviation of $11,000. Assuming the population standard deviations are equal, use the 0.05 level of significance in testing whether the population means could be equal for customers serviced by the two accounts manages. Using the appropriate statistical table, what is the most accurate statement we can make about the p-value for this test? Construct and interpret the 95% confidence interval for the difference between the population means.

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To begin with, you are trying to test whether the population means of the two groups (customers serviced by two account managers) could be equal. This involves a hypothesis test.

Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The population means are equal (μ1 = μ2)
Alternative hypothesis (Ha): The population means are not equal (μ1 ≠ μ2)

Step 2: Determine the level of significance:
Given in the question is a significance level of 0.05. This means that we are willing to accept a 5% chance of rejecting the null hypothesis even if it is true.

Step 3: Select the appropriate test statistic:
Since the sample sizes are large enough, we can use the two-sample independent t-test to compare the means of the two groups.

Step 4: Calculate the test statistic:
The formula to calculate the t-value is:
t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2))
Where:
x1 = sample mean of the first group (40 customers)
x2 = sample mean of the second group (30 customers)
s1 = standard deviation of the first group
s2 = standard deviation of the second group
n1 = sample size of the first group
n2 = sample size of the second group

In this case:
x1 = $23,000, s1 = $8500, n1 = 40 (for the first group)
x2 = $28,000, s2 = $11,000, n2 = 30 (for the second group)

Step 5: Determine the critical region:
Since the problem statement mentions that we are using a significance level of 0.05, we are conducting a two-tailed test. This means that we divide the alpha level equally into two tails, with 0.025 on each side. We will look up the critical t-value from the t-distribution table with the degrees of freedom calculated as follows: df = (n1 - 1) + (n2 - 1).

Step 6: Calculate the p-value:
The p-value can be calculated by comparing the test statistic (t-value) obtained in step 4 with the critical t-value obtained in step 5. If the test statistic falls within the critical region (extreme values), the p-value will be less than the significance level (0.05), indicating that we reject the null hypothesis.

Once the p-value is obtained, you can interpret it as follows:
- If the p-value is less than 0.05, the difference in means is statistically significant, and we reject the null hypothesis.
- If the p-value is greater than 0.05, the difference in means is not statistically significant, and we fail to reject the null hypothesis.

Regarding the second part of your question, constructing a 95% confidence interval for the difference between the population means involves the following steps:

Step 1: Calculate the standard error of the difference:
The formula to calculate the standard error of the difference between the means is:
SE = sqrt((s1^2 / n1) + (s2^2 / n2))

Step 2: Calculate the margin of error:
The margin of error (ME) can be calculated using the formula:
ME = t * SE
Where t is the critical t-value obtained from the t-distribution table for a 95% confidence interval and the degrees of freedom (df) calculated as described above.

Step 3: Calculate the confidence interval:
The confidence interval can be calculated using the formula:
CI = (x1 - x2) ± ME
Where x1 and x2 are the sample means, and ME is the margin of error.

Interpreting the confidence interval:
The confidence interval will provide a range within which we can be 95% confident that the true difference between the population means lies. If the interval contains zero, it suggests that the difference in means is not statistically significant at a 5% level of significance.

Remember, while these steps outline the process, it's essential to perform the actual calculations to obtain the precise values for the test statistic, critical t-value, p-value, standard error, margin of error, and confidence interval.