C Appendix: Symbols used for statistics and parameters

Almost all confidence intervals have the form

$$\text{statistic}\pm (\text{multiplier}\times \text{s.e.}(\text{statistic}\left)\right).$$ Notes:

• The multiplier is approximately 2 for an approximate 95% CI (based on the 68–95–99.7 rule).
• $\text{multiplier}\times \text{s.e.}\left(\text{statistic}\right)$ is called the margin of error.
• Odds ratios have a slight complication, so this formula does not apply for odds ratio (for which the test statistic is ${\chi }^2$ and not $t$). For the same reason, a standard error for ORs is not given.

For hypothesis testing, $t$-scores all have the form:

$$t=\frac{\text{statistic}-\text{parameter}}{\text{s.e.}\left(\text{statistic}\right)}$$ Notes:

• Odds ratios have a slight complication, so this formula does not apply for odds ratio (for which the test statistic is ${\chi }^2$ and not $t$). For the same reason, a standard error for ORs is not given.
• The ${\chi }^2$ statistic is approximately like a $z$-score of (where $\text{df}$ is the ‘degrees of freedom’ given in the software output):

$$\sqrt{\frac{{\chi }^2}{\text{df}}}.$$

TABLE C.1: Some sample statistics used to estimate population parameters. Empty table cells means that these are not studied. The asterisk means that no formula is given in this text.

	Parameter	Statistic	Standard error	S.E. formula reference
Proportion	$p$	$\hat{p}$	$\text{s.e.}\left(\hat{p}\right)=\sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}$	Def. 20.2
Mean	$\mu$	$\overline{x}$	$\text{s.e.}\left(\overline{x}\right)=\frac{s}{\sqrt{n}}$	Def. 22.1
Standard deviation	$\sigma$	$s$
Mean difference	${\mu }_d$	$\overline{d}$	$\text{s.e.}\left(\overline{d}\right)=\frac{s_d}{\sqrt{n}}$	Def. 23.2
Diff. between mean	${\mu }_1-{\mu }_2$	${\overline{x}}_1-{\overline{x}}_2$	$\text{s.e.}({\overline{x}}_1-{\overline{x}}_2)$
Odds ratio	Pop. OR	Sample OR	$\text{s.e.}\left(\text{sample OR}\right)$
Correlation	$\rho$	$r$
Slope of regression line	${\beta }_1$	$b_1$	$\text{s.e.}\left(b_1\right)$
Intercept of regression line	${\beta }_0$	$b_0$	$\text{s.e.}\left(b_0\right)$
R-squared		$R^2$

TABLE C.2: Some other symbols used

Symbol	Meaning	Reference
$H_0$	Null hypothesis	Sect. 28.2
$H_1$	Alternative hypothesis	Sect. 28.2
df	Degrees of freedom	Sect. 31.4
CI	Confidence interval	Chap. 21
s.e.	Standard error	Def. 18.2
$n$	Sample size
${\chi }^2$	The chi-squared test statistic	Sect. 31.4