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.

To test whether the population means could be equal for customers serviced by the two account managers, we can use a two-sample t-test. The null hypothesis (H₀) is that the population means are equal, and the alternative hypothesis (H₁) is that they are not equal.

The formula for the t-test statistic is:

t = (x₁ - x₂) / √((s₁²/n₁) + (s₂²/n₂))

Where:
x₁ = sample mean for account manager 1
x₂ = sample mean for account manager 2
s₁ = standard deviation for account manager 1
s₂ = standard deviation for account manager 2
n₁ = sample size for account manager 1
n₂ = sample size for account manager 2

Plugging in the given values:
x₁ = $23,000, x₂ = $28,000
s₁ = $8,500, s₂ = $11,000
n₁ = 40, n₂ = 30

Calculating the t-value:
t = ($23,000 - $28,000) / √(($8,500²/40) + ($11,000²/30))

After calculating the t-value, we can compare it to the critical value from the t-distribution table with (n₁ + n₂ - 2) degrees of freedom and a significance level of 0.05. If the t-value is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject it.

To construct the 95% confidence interval for the difference between the population means, we can use the formula:

CI = (x₁ - x₂) ± t_critical * √((s₁²/n₁) + (s₂²/n₂))

Where t_critical is the critical value from the t-distribution table with (n₁ + n₂ - 2) degrees of freedom and a significance level of 0.05. By substituting the given values, we can calculate the lower and upper bounds of the confidence interval.

For the p-value, we analyze its relationship with the significance level. If the p-value is less than or equal to the significance level (0.05 in this case), we reject the null hypothesis. If it is greater, we fail to reject the null hypothesis.

To determine the most accurate statement about the p-value, we need to compare it to different significance levels. If the p-value is less than 0.01, we can state that the result is highly significant. If it is between 0.01 and 0.05, we can say it is significant. If it is greater than 0.05, we can conclude that it is not significant.

Therefore, without calculating the specific values, we can only state the possible conclusions about the p-value based on the given significance level of 0.05. We can say that the most accurate statement about the p-value for this test is either "significant" or "not significant" based on whether it is less than or equal to 0.05 or greater than 0.05, respectively.