# C Symbols, formulas, statistics and parameters

## C.1 Symbols and standard errors

Parameter | Statistic | Standard error | S.E. formula reference | |
---|---|---|---|---|

Proportion | \(p\) | \(\hat{p}\) | \(\displaystyle\text{s.e.}(\hat{p}) = \sqrt{\frac{ \hat{p} \times (1 - \hat{p})}{n}}\) | Def. 20.2 |

Mean | \(\mu\) | \(\bar{x}\) | \(\displaystyle\text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}}\) | Def. 22.1 |

Standard deviation | \(\sigma\) | \(s\) | ||

Mean difference | \(\mu_d\) | \(\bar{d}\) | \(\displaystyle\text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n}}\) | Def. 23.2 |

Diff. between means | \(\mu_1 - \mu_2\) | \(\bar{x}_1 - \bar{x}_2\) | \(\displaystyle\text{s.e.}(\bar{x}_1 - \bar{x}_2)\) | -- |

Odds ratio | Pop. OR | Sample OR | \(\displaystyle\text{s.e.}(\text{sample OR})\) | -- |

Correlation | \(\rho\) | \(r\) | ||

Slope of regression line | \(\beta_1\) | \(b_1\) | \(\text{s.e.}(b_1)\) | -- |

Intercept of regression line | \(\beta_0\) | \(b_0\) | \(\text{s.e.}(b_0)\) | -- |

R-squared | \(R^2\) |

## C.2 Confidence intervals

Almost all **confidence intervals** have the form

\[ \text{statistic} \pm ( \text{multiplier} \times \text{s.e.}(\text{statistic})). \]

**Notes:**

- The multiplier is
*approximately*2 for an*approximate*95% CI (based on the 68--95--99.7 rule). -
\(\text{multiplier} \times \text{s.e.}(\text{statistic})\) is called the
*margin of error*. - Confidence intervals for
*odds ratios*are slightly different, so**this formula does not apply for odds ratios**. For the same reason, a standard error for ORs is not given.

## C.3 Hypothesis testing

For many **hypothesis tests**, the *test statistic* is a \(t\)-score, which has the form:

\[ t = \frac{\text{statistic} - \text{parameter}}{\text{s.e.}(\text{statistic})}. \]

**Notes:**

- Since \(t\)-scores are a little like \(z\)-scores, the 68--95--99.7 rule can be used to
*approximate*\(P\)-values. - Tests involving
*odds ratios*do not use \(t\)-scores, so**this formula does not apply for tests involving odds ratios**. - For tests involving odds ratios, the
*test statistic*is a \(\chi^2\) score and not \(t\)-score. For the same reason, a standard error for ORs is not given. - The \(\chi^2\) statistic is approximately like a \(z\)-score with a value of (where \(\text{df}\) is the 'degrees of freedom' given in the software output):

\[ \sqrt{\frac{\chi^2}{\text{df}}}. \]

## C.4 Other formulas

- To estimate the sample size needed when
**estimating a proportion**: \(\displaystyle n = \frac{1}{(\text{Margin of error})^2}\). - To estimate the sample size needed when
**estimating a mean**: \(\displaystyle n = \left( \frac{2\times s}{\text{Margin of error}}\right)^2\). - To calculate \(z\)-scores: \(\displaystyle z = \frac{x - \mu}{\sigma}\) or, more generally, \(\displaystyle z = \frac{\text{specific value of variable} - \text{mean of variable}}{\text{measure of variable's variation}}\).
- The
**unstandardizing formula**: \(x = \mu + (z\times \sigma)\).

**Notes:**

- In
**sample size calculations**, always**round up**the sample size found from the above formulas.