### 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case

*s* = the standard deviation of sample values.

$t=\frac{\stackrel{-}{x}-\mu}{\frac{s}{\sqrt{n}}}$ 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: $\stackrel{-}{x}-{t}_{\mathrm{v,\alpha}}\left(\frac{s}{\sqrt{n}}\right)\le \mu \le \stackrel{-}{x}+{t}_{\mathrm{v,\alpha}}\left(\frac{s}{\sqrt{n}}\right)$

### 8.3 A Confidence Interval for A Population Proportion

*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′* 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.

### 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables

*n* = $\frac{{Z}^{2}{\sigma}^{2}}{(\stackrel{-}{x}-\mu {)}^{2}}$ = 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