Based on z-score, is it possible to compute confidence without looking at a z-table? How are the values in a z-table computed? Is it possible to compute confidence without looking up a z-table?
 A: A z-table gives you values of the cumulative distribution function for the standard (i.e., mean = 0, standard deviation = 1) normal distribution. The integral needs to be estimated numerically. See here for a discussion.
Of course, to answer your question literally, you could use some other lookup system other than a table, such as the pnorm function in R.
Example:
> pnorm(2)
[1] 0.9772499

Help for pnorm also provides the following reference on how to calculate the cdf of a normal distribution:


*

*Cody, W. D. (1993) Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers. ACM Transactions on Mathematical Software 19, 22–32.

A: The simplest answer is that one either uses numerical quadrature techniques (Simpson's, for instance) on the PDF, or approximations such as those found in Abramowitz and Stegun. Personally however, since one merely needs a few digits of accuracy for the "error function" that is involved in computing z-scores, you might be interested the simple approximations given here by S. Winitzki.
A: 
How are the values in a z-table computed?

You can use more than one methods. You can use the Simpson's rule and get an approximate a single individal value in both the positive or negative side of the standard normal table. You can also use series approximation or numerical integration.

Is it possible to compute confidence without looking up a z-table?

You would be more likely to call a function in a package than look up a table. When needed to judge significance of a Z test, a lot of statisticians will know two tailed 20% 10%, 5% & 1% critical values by heart.
Disc: I'm affiliated with the site link above
