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 manager averaged $\$ 28,000$ in transactions, 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 account managers. 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.

Solution

VerifiedLet's define the following:

- $n_1=40$ is the size of a sample $1$,
- $n_2=30$ is the size of a sample $2$,
- $\overline{x}_1=23000$ is a mean of sample $1$,
- $\overline{x}_2=28000$ is a mean of sample $2$,
- $s_1=8500$ is a standard deviation of sample $1$,
- $s_2=11000$ is a standard deviation of sample $2$.