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

*s* = the standard deviation of sample values.

$t=\frac{\stackrel{-}{x}\xe2\u02c6\u2019\mathrm{\xce\xbc}}{\frac{s}{\sqrt{n}}}$ is the formula for the *t*-score which measures how far away a measure is from the population mean in the Studentâ€™s t-distribution

*df* = *n* - 1; the degrees of freedom for a Studentâ€™s t-distribution where n represents the size of the sample

*T*~*t _{df}* the random variable,

*T*, has a Studentâ€™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: $\stackrel{-}{x}-{t}_{\mathrm{v,\xce\pm}}\left(\frac{s}{\sqrt{n}}\right)\xe2\u2030\xa4\mathrm{\xce\xbc}\xe2\u2030\xa4\stackrel{-}{x}+{t}_{\mathrm{v,\xce\pm}}\left(\frac{s}{\sqrt{n}}\right)$

### 8.3 A Confidence Interval for A Population Proportion

*pâ€²= * $\frac{x}{n}$ where *x* represents the number of successes in a sample and *n* represents the sample size. The variable *p*â€² is the sample proportion and serves as the point estimate for the true population proportion.

*q*â€² = 1 â€“ *p*â€²

The variable *pâ€²* 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:

$\mathrm{p\text{'}}-{Z}_{\mathrm{\xce\pm}}\sqrt{\frac{\mathrm{p\text{'}q\text{'}}}{n}}\xe2\u2030\xa4p\xe2\u2030\xa4\mathrm{p\text{'}}+{Z}_{\mathrm{\xce\pm}}\sqrt{\frac{\mathrm{p\text{'}q\text{'}}}{n}}$

$n=\frac{{Z}_{\frac{\mathrm{\xce\pm}}{2}}{}^{2}{p}^{\xe2\u20ac\xb2}{q}^{\xe2\u20ac\xb2}}{{e}^{2}}$ provides the number of observations needed to sample to estimate the population proportion, *p*, with confidence 1 - *Î±* 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* = $\frac{{Z}^{2}{\mathrm{\xcf\u0192}}^{2}}{(\stackrel{-}{x}-\mathrm{\xce\xbc}{)}^{2}}$ = 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

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