This is how I would have done this. Since it's a sample, you need to find what I call . (p-bar is the same as p-hat, and I don't know how to do the symbol for that either ;) and that doesn't look how I mean -- i.e. sigma sub-p-bar) Your book could call this something else. This is the standard deviation for a sample proportion.

The equation I know for this is:



I seem to recall there's another equation for this, but don't think I ever learned it. Again, that needs to be looked up in the text to see which one is being used.

Using this equation, I get:



From there, you can use the same z equation. If you think of the z equation, rather than symbols, as:




By says "its points" and "its standard deviation" I mean the ones that belong to the point you're using. i.e. is your point an x, an x-bar, or a p-bar? This can be applied whether you're doing a population, a sample mean, or a proportion. They all work exactly the same way. In this case, the point is the .5 you're finding the probability of, "its mean" is .6, and "its standard deviation" is the .0586 I figured out.

As for whether it could be a t, different books have different tests for this. The one I learned is np > .05 and np(1-p) > .05. Again, got to find that in the particular text being used.

Unfortunately, much of this is book-dependent. But this is the basic gist of it.