Please can you give the formula 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.
To test whether the population means could be equal for customers serviced by the two account managers, we need to perform a two-sample t-test. The formula for calculating the t-statistic for two independent samples with equal population standard deviations is given by:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where:
- x1 and x2 are the sample means (average transactions) for the two account managers
- s1 and s2 are the sample standard deviations for the two account managers
- n1 and n2 are the sample sizes for the two account managers
For the given data:
x1 = $23,000, x2 = $28,000
s1 = $8,500, s2 = $11,000
n1 = 40, n2 = 30
Substituting these values in the formula, we get:
t = (23,000 - 28,000) / sqrt((8,500^2 / 40) + (11,000^2 / 30))
Now we need to find the degrees of freedom (df) for the t-distribution, which is calculated as:
df = (s1^2 / n1 + s2^2 / n2)^2 / [(s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1)]
Substituting the values:
df = (8,500^2 / 40 + 11,000^2 / 30)^2 / [(8,500^2 / 40)^2 / (40 - 1) + (11,000^2 / 30)^2 / (30 - 1)]
Now, we can use the t-distribution table or a statistical software to look up the critical t-value at a 0.05 level of significance (alpha = 0.05) with the calculated degrees of freedom (df). Let's assume it is t_critical.
If the calculated t-value (t) is greater than t_critical, we reject the null hypothesis (population means are equal) in favor of the alternative hypothesis (population means are not equal).
Now moving on to the construction and interpretation of the 95% confidence interval for the difference between the population means:
The formula for calculating the confidence interval (CI) is given by:
CI = (x1 - x2) ± t_critical * sqrt((s1^2 / n1) + (s2^2 / n2))
Substituting the values:
CI = (23,000 - 28,000) ± t_critical * sqrt((8,500^2 / 40) + (11,000^2 / 30))
By calculating the upper and lower bounds of the confidence interval, we can interpret it as follows:
"We are 95% confident that the true difference between the population means of customers serviced by the two account managers falls within the calculated confidence interval."
Please note that I have provided the general steps and explanation for solving the problem, but the specific values for t_critical and the confidence interval need to be determined based on the actual calculations or using statistical software.