A salesperson promoting an ultrasound exam machine from $2013$ claims that it can significantly reduce the time required for exams. However, the procurement department of USC Hospital is uncertain about the reliability of this assertion. Fortunately, the hospital has the capability to monitor the actual exam times for each ultrasound machine, as they are recorded in a file named "ultra$\_$sound.csv$".$ Additionally, the hospital can track the exam times for both machine types and distinguish between practitioners based on their experience level. This information will enable the hospital to determine whether it should replace its current models from $2008$ with the new $2013$ machines. It is important to note that the hospital will only consider replacing the $2008$ machine with the $2013$ model if the latter demonstrates shorter exam times.

$1.$ Run the following regression and report the followings:

Exam Time! $=$ b$"+$ b#New Machine! $+$ e$!$

a$.$ Report the estimates of b$"$ and b# with $-$values and p$-$values.

b$.$ What is R$2?$

c$.$ What is the estimated standard error of e$!?$

d$.$ This regression analysis aims to determine whether it is necessary for the hospital to replace their

old ultrasound model with the new $2013$ machines, by investigating whether there is a significant reduction in exam time for the new machines compared to the old ones. What is your conclusion based on the estimation results?

$2.$ Run the following regression and report the followings:

Exam Time! $=$ b$"+$ b#New Machine! $+$ b$Expert! $+$ e$!$

a$.$ Report the estimates of b$",$ b# and b$ with $-$values and $-$values.

b$.$ What is R$2?$

c$.$ Based on this regression, does hospital replace their old model with the new $2013$ ultra sound

machines? In other words, is the exam time significantly lower for the new machines than old machines?

$3.$ Do the conclusions reached in question $1.$d and question $2.$c differ? If they do$,$ can you explain why?

Hint: Complete the table provided below.

Average Exam Time

New machine Expert Novice Old machine Expert Novice

$4.$ Run the following regression and report the followings:

Number of Observations

Exam Time! $=$ b$"+$ b#New Machine! $+$ b$Expert! $+$ b$\%$New Machine! $$ Expert! $+$ e$!$

a$.$ Report the estimates of b#$,$ b$$,$ and b$\%$ with $-$values and $-$values. b$.$ Provide the interpretation for the estimates b#$,$ b$$,$ and b$\%.$

$1$

Definition of the following two variables

Appendix

$$h$$&

$-$ A dummy variable which is $1,$ if the observation $$th machine is $$new$\_2013$ and $0$ otherwise.

$$&

$-$ A dummy variable which is $1,$ if the observation $$th expertise is $$expert$$ and $0$ otherwise.