A study is conducted in patient with HIV. Lower CD4 count is associated with advance disease. The researcher is interested in finding out the relation between supplement vitamins and CD4 count. A multiple regression analysis is performed relating CD4 count to the use of suppliments(coded as 1=yes, 0=No)and to the duration of HIV in years between the diagnosis of HIV and the study dates). For the analysis,

y=CD4count
Y hat=501.41+12.67(supplement)-30.23(duration of HIV).
a) what is expected CD4 countfor a patient taking supplement who has had HIV for 2.5 years?
b)what is the CD4 count For patient not taking supplement who was diagnosed with HIV at study enrollment?
c)what is the CD4 count for patients not taking supplement who has had HIV for 2.5 years?

To answer these questions, we will use the multiple regression equation given:

Y hat = 501.41 + 12.67(supplement) - 30.23(duration of HIV)

a) For a patient taking supplements who has had HIV for 2.5 years, we need to plug in the values into the equation. Since the patient is taking supplements, the supplement value would be 1, and since the duration of HIV is 2.5 years, the duration value would be 2.5. So we can calculate the expected CD4 count:

Y hat = 501.41 + 12.67(1) - 30.23(2.5)
Y hat = 501.41 + 12.67 - 75.575
Y hat = 438.505

Therefore, the expected CD4 count for a patient taking supplements who has had HIV for 2.5 years is approximately 438.505.

b) For a patient not taking supplements who was diagnosed with HIV at study enrollment, we again need to plug in the values into the equation. Since the patient is not taking supplements, the supplement value would be 0, and since the duration of HIV is not mentioned, we cannot calculate the exact value. We can assume a duration value of 0 for study enrollment since the duration is between the diagnosis and the study dates. So we can calculate the CD4 count:

Y hat = 501.41 + 12.67(0) - 30.23(0)
Y hat = 501.41 + 0 - 0
Y hat = 501.41

Therefore, the CD4 count for a patient not taking supplements who was diagnosed with HIV at study enrollment is 501.41.

c) For patients not taking supplements who have had HIV for 2.5 years, we again need to plug in the values into the equation. Since the patient is not taking supplements, the supplement value would be 0, and the duration value would be 2.5. So we can calculate the CD4 count:

Y hat = 501.41 + 12.67(0) - 30.23(2.5)
Y hat = 501.41 + 0 - 75.575
Y hat = 425.835

Therefore, the CD4 count for patients not taking supplements who have had HIV for 2.5 years is approximately 425.835.