Working off Peter Flom's suggestion, look at this R code:
x <- as.data.frame(matrix(rnorm(50000,0,1), nr = 5000, nc = 10))
x$y <- x$V1 + x$V2 + x$V3 + x$V4 + x$V5 + x$V6 + x$V7 + x$V8 + x$V9 + x$V10
We generate 10 variables (V1 through V10), each with a mean of 0 and a standard deviation of 1. Y is constructed to be the sum of the ten variables for each observation. You can then run:
And those results should give you some insight about what to expect.
Then, look at this Wikipedia entry (the formula for Z specifically) and see if your insights match theirs!