Confidence Interval Formulas We are doing the topic confidence interval in class and I just wanted to confirm if my concept about it was clear.
Single Population:
$x \pm Z*$ $\frac{\sigma}{\sqrt{n}}$ Now this is used when we know the population variance or we don't know the population variance but $n > 30$ (We would replace $\sigma$ by $s$.
$x \pm t*$ $\frac{s}{\sqrt{n}}$  This is the formula we use when we don't know the population variance and $n \leq 30$ (We can also apply this when $ n > 30 $ too).
$p \pm Z*$ $\sqrt{\frac{p(1-p)}{n}}$ Now I know we apply this when proportions are involved.
2 Populations:
$x_1 - x_2 \pm Z*$ $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} $ This is when we know the population variance of both the populations.
$x_1 - x_2 \pm Z*$ $\sqrt{\frac{s^2}{n_1} + \frac{s^2}{n_2}} $ This is the formula when we do not know the population variance of any of the population but $ n_1 + n_2 > 30 $.
$x_1 - x_2 \pm t*s_p*$ $\sqrt{\frac{1}{n_1} + \frac{1}{n_2}} $ This is when we do not the population variance of any of the populations and $n_1 + n_2 <30 $.
Plus I wanted to know the formula for proportions in the case of two populations.
What I want to know is if I'm applying the correct formula in each case. Plus are my formulas correct. (My professor has a habit of sometimes missing a constant etc. when telling us about the formulas.)
Thanks in advance :)
 A: ($\alpha$ is size of a test)
Single Population:
$\bar x \pm Z_{\alpha/2}*$ $\frac{\sigma}{\sqrt{n}}$ this is used when we know the population variance but $n > 30$(large n is needed, some say $n>20$ some say $n>30$), We would replace $\sigma$ by $S$ if you apply Central Limit Theorem.
$\bar x \pm t_{n-1,\alpha/2}*$ $\frac{s}{\sqrt{n}}$  This is the formula we use when we don't know the population variance and as $n\rightarrow\infty~,~t_n\rightarrow N(0,1)$
$p \pm Z_{\alpha/2}*$ $\sqrt{\frac{p(1-p)}{n}}$ we apply this when proportions are involved with occurrence are 0,1.
2 Populations:
$\bar x_1 - \bar x_2 \pm Z_{\alpha/2}*$ $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} $  when we know the population variance of both the populations.
$\bar x_1 - \bar x_2 \pm t_{n_1+n_2-2,\alpha/2}*s_p*$ $\sqrt{\frac{1}{n_1} + \frac{1}{n_2}} $ when we do not the population variance of any of the populations and as $n\rightarrow\infty~,~t_n\rightarrow N(0,1)$
$p_1-p_2 \pm Z_{\alpha/2}*$ $\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$ we apply this when proportions are involved with occurrence are 0,1
