z.test or t.test when n is large, but variance unknown? I am trying to decide whether to do a t.test or a z.test in R to check the hypothesis if two samples have the same mean, $\mu_1 = \mu_2$ with a 95% certainty. 
What I know is that we use t.test if $\sigma^2$ is unknown and should be approximated. In my case the samples sizes are $S_1 = 1000$ and $S_2 = 1000$. So, I have quite a big sample, and if I check the t-values when $df = 999$ and $\alpha = 0.025$ they have approximately the same values as the z-values, from a $\mathcal{N}(0,1)$. 
So according to the central limit theorm, can I do a z.test and assume $\sigma^2 = 1$, or should I stick to the t.test with the ridiculously degree of freedom size? 
Since $t \sim(\infty) = \mathcal{N}(0,1) $
Can I assume $\sigma^2 = 1$ ?  
 A: The quick answer: you should do a t-test, rather than a z-test (although you are correct in that the results will be nearly identical). 
The long answer: yes, it is true that as $df \rightarrow \infty$, $t_{df} \rightarrow_D N(0,1)$, i.e. that as the degrees of freedom approach infinity, a t distribution approaches a normal distribution. Historically, this was very useful, as most statisticians did not have access to the table of quantiles for, say, a $t_{5023}$ distribution. But with modern computers, this is not a problem at all. You are correct in that there would be very little inaccuracy in using a z-test instead of a t-test with $df = 999$. But given that nowadays, it is just as easy to use a t-test as a z-test, there is no reason not to. 
Also, I may be misinterpreting what you said, but I wanted to clear up some confusion just in case. In the z-test, we have 
$\frac{\bar x_1 - \bar x_2}{ \sqrt{ \sigma_1^2/n_1 + \sigma_2^2/n_2 } } \sim N(0,1)$ under the null hypothesis.
So if you wanted to approximate a t-test with a z-test, you would use $s_1$ and $s_2$ for $\sigma_1$ and $\sigma_2$, not $\sigma = 1$. But perhaps I'm a little confused on what you meant when you said "use $\sigma = 1$", i.e. this would be the standard deviation of the reference distribution.
