C Appendix: Symbols, formulas, statistics and parameters
Symbols used
Symbol | Meaning | Reference |
---|---|---|
H0 | Null hypothesis | Sect. 28.2 |
H1 | Alternative hypothesis | Sect. 28.2 |
df | Degrees of freedom | Sect. 31.4 |
CI | Confidence interval | Chap. 21 |
s.e. | Standard error | Def. 18.3 |
n | Sample size | |
χ2 | The chi-squared test statistic | Sect. 31.4 |
Confidence intervals
Almost all confidence intervals have the form
statistic±(multiplier×s.e.(statistic)). Notes:
- The multiplier is approximately 2 for an approximate 95% CI (based on the 68–95–99.7 rule).
- multiplier×s.e.(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.
Hypothesis testing
For many hypothesis tests, the test statistic is a t-scores, which has the form:
t=statistic−parameters.e.(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 χ2 score and not t-score. For the same reason, a standard error for ORs is not given.
- The χ2 statistic is approximately like a z-score with a value of (where df is the ‘degrees of freedom’ given in the software output):
√χ2df.
Parameter | Statistic | Standard error | S.E. formula reference | |
---|---|---|---|---|
Proportion | p | ˆp | s.e.(ˆp)=√ˆp×(1−ˆp)n | Def. 20.2 |
Mean | μ | ˉx | s.e.(ˉx)=s√n | Def. 22.1 |
Standard deviation | σ | s | ||
Mean difference | μd | ˉd | s.e.(ˉd)=sd√n | Def. 23.2 |
Diff. between mean | μ1−μ2 | ˉx1−ˉx2 | s.e.(ˉx1−ˉx2) |
|
Odds ratio | Pop. OR | Sample OR | s.e.(sample OR) |
|
Correlation | ρ | r | ||
Slope of regression line | β1 | b1 | s.e.(b1) |
|
Intercept of regression line | β0 | b0 | s.e.(b0) |
|
R-squared | R2 |