When computing the confidence interval for the population proportion, the Student's t-distribution is used rather than the normal distribution.

As the confidence level for a confidence interval increases, the width of the interval also increases.

A $$100\left(1-\alpha\right)\%$$  confidence interval for the difference between two means, independent samples, and known population variances is given by:

$$\overline{x}-\overline{y}\pm z_{\frac{\alpha}{2}}\sqrt[]{\frac{\sigma_x^2}{n_x}-\frac{\sigma^2y}{n_y}}$$  

Interval estimates for the variance of a normal population rely on the random variable $$\frac{\left(n-1\right)s^2}{\sigma^2}$$ which follows

According to the central limit theorem, the sampling distribution of sample means can be approximated by the normal distribution as the number of samples gets large enough.