Is the sample independent from the mean? I came across this question in Dobson's Introduction to Generalized Linear Models: 
What concerns me is that statement that the $\bar{Y}_j$ and $Y_{jk}$ are independent. How can $\bar{Y}_j$ (the sample mean for a particular group $j$) be independent with all the $Y_{jk}$ (all the observations in the sample)?

This exercise is about an unpaired t-test (based on the instructions directly above the image). From a previous page:

"Let $Y_jk$ denote the kth observation in the jth group where j = 1
for the treatment group, j = 2 for the control group and k = 1, . . . , 20 for both groups." (p. 44)

So here, there are two $\overline{Y}_j$ one for each group. Because this is an independent t-test, the group means should be independent of one another. And each j is independent in this example, so the random k observations in group j=1 should be independent of the random k observations in group j = 2.
