Total patients = 2530
Total patients needing 1 dose = 70%
Total patients needing 2-3 doses = 30%
What is the average number of doses needed by the total 2530 patients?
[MATH]\overline{D}=0.7(1)+0.3\left(\frac{2+3}{2}\right)=1.45[/MATH]
It's worth pointing out that this (though clearly what they consider "correct", and really the only thing you can do) is really a guess. We are supposing that the "patients needing 2-3 doses" are evenly divided between 2 and 3 doses, so their average is 2.5 doses each.
Another way to see the same thing is to split the "2-3" into equal parts:
Total patients = 2530
Total patients needing 1 dose = 70%
Total patients needing 2 doses = 15%
Total patients needing 3 doses = 15%
We don't know this is how things really are; but we have to assume something in order to give an answer.
Now, if you aren't convinced about the weighted average, you could just imagine listing the number of doses each patient gets, and adding them all up:
Total patients = 2530
Total patients needing 1 dose = 70%*2530 = 1771
Total patients needing 2 doses = 15%*2530 = 379.5
Total patients needing 3 doses = 15%*2530 = 379.5
Then the total number of doses is 1771*1 + 379.5*2 + 379.5*3 = 3668.5. Dividing that by the number of patients, the average number of doses is 3668.5/2530 = 1.45.
Of course, the weighted average method is quicker -- that's why it was invented! It also shows that it doesn't matter how many patients there are -- unless you care about the fact that we assumed a couple half-patients!
