Sample size when adding two means

I want to compare a theoretical mean to a measured mean. But I determine this theoretical mean from adding two measured means. For example, let's say I want to determine if mixing a shade of white paint with a shade of black paint yields a particular gray paint (let’s pretend I can quantify paint color).

I find the mean hexadecimal color of the white paint by measuring several independent samples of Eggshell white (n=20 different paint cans) and I do the same for the Midnight black paint (n=28), lets pretend these all come from different companies so there is variation between each paint can of the same color. I also have a gray paint that I measure the color of using, again, several samples (n=28).

I want to then ask if I were to theoretically mix the white and black paint (based on the individual samples I measured) would they add up to the gray paint's measured color? In other words ask: If I add the mean color of the white paint, to the mean color of the black paint would this theoretical value differ from my actual measurements of the gray paint’s color?

So I have the sample size, mean, and std dev of the gray paint I measured. I have the mean of the theoretical gray paint (found by adding the means of white and black paint) and the std dev of the theoretical gray paint (found by quadrating the std dev of the white and black samples). But what I am missing is the sample size of the theoretical gray paint, because I cannot simply add the white and black samples sizes together, as they are different colors.

I need to perform a t-test to ask is my measured gray paint significantly different than the theoretical gray paint? My hunch would be to use the smaller sample size of the two paints (n=20) as a conservative value but I'm unsure if this is appropriate, because technically this theoretical value does not have a number of samples associated with it. Or perhaps a t-test isn't appropriate to determine if there is a significant difference between the two?


Well-Known Member
Hi MH,

I will try to answer the question only.

you have a random variable X=X1+X2
you know the theoretical parameters of X1 and X2, now you can calculate the theoretical parameters of X.

Now you have a new question.
You know the theoretical parameters of X, you take a sample and you want to compare the theoretical mean to the measured mean.
I assume you know how to run this test in?