s = the standard deviation of sample values.

$t= \frac{\bar{x}-\mu }{\frac{s}{\sqrt{n}}}$ is the formula for the t-score which measures how far away a measure is from the population mean in the Student’s t-distribution

df = n - 1; the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

T~t_df the random variable, T, has a Student’s t-distribution with df degrees of freedom

The general form for a confidence interval for a single mean, population standard deviation unknown, and sample size less than 30 Student's t is given by: $\overline{x}-t_{\mathrm{v,\alpha }}\left(\frac{s}{\sqrt{n}}\right)\le \mu \le \overline{x}+t_{\mathrm{v,\alpha }}\left(\frac{s}{\sqrt{n}}\right)$

p′= $\frac{x}{n}$ where x represents the number of successes in a sample and n represents the sample size. The variable p′ is the sample proportion and serves as the point estimate for the true population proportion.

q′ = 1 – p′

The variable $p^{\prime }$ has a binomial distribution that can be approximated with the normal distribution shown here. The confidence interval for the true population proportion is given by the formula:

$\mathrm{p\text{'}}-Z_{\alpha }\sqrt{\frac{\mathrm{p\text{'}q\text{'}}}{n}}\le p\le \mathrm{p\text{'}}+Z_{\alpha }\sqrt{\frac{\mathrm{p\text{'}q\text{'}}}{n}}$

$n= \frac{Z_{\frac{\alpha }{2}}{}^2{p}^{\prime }{q}^{\prime }}{e^2}$ provides the number of observations needed to sample to estimate the population proportion, p, with confidence 1 - α and margin of error e. Where e = the acceptable difference between the actual population proportion and the sample proportion.

n = $\frac{Z^2{\sigma }^2}{(\overline{x}-\mu {)}^2}$ = the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence for a continuous random variable

$n=\frac{Z_{\alpha }^2\mathrm{pq}}{e^2}$ = the formula used to determine the sample size if the random variable is binary