C Symbols, formulas, statistics and parameters
C.1 Symbols and standard errors
The following table lists the statistics used to estimate unknown population parameters.
The sampling distribution is given for each statistic, where possible.
When the sampling distribution is approximately normally distributed, under appropriate statistical validity conditions, this is indicated by .
-
The value of the mean of the sampling distribution (the sampling mean) is:
- unknown, for confidence intervals.
- assumed to be the value given in the null hypothesis, for hypothesis tests.
| Statistic | sampling mean | distn? | error | Ref. | |
|---|---|---|---|---|---|
| Proportion | \(\hat{p}\) | \(p\) | ✔ | CI: \(\displaystyle \sqrt{\frac{ \hat{p} \times (1 - \hat{p})}{n}}\) | Ch. 22 |
| ✔ | HT: \(\displaystyle \sqrt{\frac{ p \times (1 - p)}{n}}\) | Ch. 26 | |||
| Mean | \(\bar{x}\) | \(\mu\) | ✔ | \(\displaystyle \frac{s}{\sqrt{n}}\) | Chs. 23, 27 |
| Mean difference | \(\bar{d}\) | \(\mu_d\) | ✔ | \(\displaystyle \frac{s_d}{\sqrt{n}}\) | Ch. 29 |
| Difference between means | \(\bar{x}_1 - \bar{x}_2\) | \(\mu_1 - \mu_2\) | ✔ | \(\displaystyle \sqrt{\text{s.e.}(\bar{x}_1) + \text{s.e.}(\bar{x}_2)}\) | Ch. 30 |
| Difference between proportions | \(\hat{p}_1 - \hat{p}_2\) | \(p_1 - p_2\) | ✔ | CI: \(\displaystyle \sqrt{\text{s.e.}(\hat{p}_1) + \text{s.e.}(\hat{p}_2)}\) | Ch. 31 |
| ✔ | Ch. 31 | ||||
| Odds ratio | Sample OR | Pop. OR | ✘ | (Not given) | Ch. 31 |
| Correlation | \(r\) | \(\rho\) | ✘ | (Not given) | Ch. 33 |
| Regression: slope | \(b_1\) | \(\beta_1\) | ✔ | \(\text{s.e.}(b_1)\) (value from software) | Ch. 33 |
| Regression: intercept | \(b_0\) | \(\beta_0\) | ✔ | \(\text{s.e.}(b_0)\) (value from software) | Ch. 33 |
C.2 Confidence intervals
For statistics whose sampling distribution has an approximate normal distribution, confidence intervals have the form \[ \text{statistic} \pm \big( \text{multiplier} \times \text{s.e.}(\text{statistic})\big). \]
Notes:
- The multiplier is approximately \(2\) to create an approximate \(95\)% CI (based on the \(68\)--\(95\)--\(99.7\) rule).
- The quantity '\(\text{multiplier} \times \text{s.e.}(\text{statistic})\)' is called the margin of error.
- Software uses exact multipliers to form exact confidence intervals.
- When the sampling distribution for the statistic does not have an approximate normal distribution (e.g., for odds ratios and correlation coefficients), this formula does not apply and the CIs are taken directly from software output when available.
C.3 Hypothesis testing
For statistics that have an approximate normal distribution, the test statistic has the form: \[ \text{test statistic} = \frac{\text{statistic} - \text{parameter}}{\text{s.e.}(\text{statistic})}, \] where \(\text{s.e.}(\text{statistic})\) is the standard error of the statistic. The test-statistic is a \(t\)-score for most hypothesis tests in this book when the sampling distribution is described by a normal distribution, but is a \(z\)-score for a hypothesis test involving one or two proportions.
Notes:
- If the test-statistic is a \(z\)-score, the \(P\)-value can be found using tables (Appendix B.1), or approximated using the \(68\)--\(95\)--\(99.7\) rule.
- If the test-statistic is a \(t\)-score, the \(P\)-value can be approximated using tables (Appendix B.1), or approximated using the \(68\)--\(95\)--\(99.7\) rule (since \(t\)-scores are similar to \(z\)-scores; Sect. 28.4).
- When the sampling distribution for the statistic does not have an approximate normal distribution (e.g., for odds ratios and correlation coefficients), this formula does not apply and \(P\)-values are taken from software when available.
- A hypothesis test about odds ratios uses a \(\chi^2\) test statistic. For \(2\times 2\) tables only, the \(\chi^2\)-value is equivalent to a \(z\)-score with a value of \(\sqrt{\chi^2}\).
C.4 Sample size estimation
All the following formulas compute the approximate minimum (i.e., conservative) sample size needed to produce a \(95\)% confidence interval with a specified margin of error (i.e., the 'give-or-take' amount).
To estimate the sample size needed for estimating a proportion (Sect. 32.3): \[ n = \frac{1}{(\text{Margin of error})^2}. \]
To estimate the sample size needed for estimating a mean (Sect. 32.4): \[ n = \left( \frac{2\times s}{\text{Margin of error}}\right)^2 \] for some estimate \(s\) of the standard deviation of the data.
To estimate the sample size needed for estimating a mean difference (Sect. 32.5): \[ n = \left( \frac{2 \times s_d}{\text{Margin of error}}\right)^2 \] for some estimate \(s_d\) of the standard deviation of the differences.
-
To estimate the sample size needed for estimating the difference between two means (Sect. 32.6): \[ n = 2\times \left( \frac{2 \times s}{\text{Margin of error}}\right)^2 \] for each group being compared, where \(s\) is an estimate of the common standard deviation in the population for both groups. This formula assumes:
- the sample size in each group is the same; and
- the standard deviation in each group is the same.
To estimate the sample size needed for estimating the difference between two proportions (Sect. 32.7): \[ n = \frac{2}{(\text{Margin of error})^2} \] for each group being compared. This formula assumes the sample size in each group is the same.
Notes:
- In sample size calculations, round up the sample size found from the above formulas.
C.5 Other formulas
- To calculate \(z\)-scores (Sect. 20.4), use \[ z = \frac{\text{value of variable} - \text{mean of the distribution of the variable}}{\text{standard deviation of the distribution of the variable}}. \] \(t\)-scores are like \(z\)-scores. When the 'variable' is a sample estimate (such as \(\bar{x}\)), the 'standard deviation of the distribution' is a standard error (such as \(\text{s.e.}(\bar{x})\)).
- The unstandardising formula (Sect. 20.8) is \(x = \mu + (z\times \sigma)\).
- The interquartile range (IQR) is \(Q_3 - Q_1\), where \(Q_1\) and \(Q_3\) are the first and third quartiles respectively (or, equivalently, the \(25\)th and \(75\)th percentiles).
- The smallest expected value (for assessing statistical validity when forming CIs and conducting hypothesis tests with proportions or odds ratio) is \[ \frac{(\text{Smallest row total})\times(\text{Smallest column total})}{\text{Overall total}}. \]
- The regression equation in the sample is \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the sample intercept and \(b_1\) is the sample slope.
- The regression equation in the population is \(\hat{y} = \beta_0 + \beta_1 x\), where \(\beta_0\) is the intercept and \(\beta_1\) is the slope.
C.6 Other symbols and abbreviations used
| Symbol | Meaning | Reference |
|---|---|---|
| RQ | Research question | Chap. 2 |
| \(s\) | Sample standard deviation | Sect. 11.7.2 |
| \(\sigma\) | Population standard deviation | Sect. 11.7.2 |
| \(s_d\) | Sample standard deviation of differences | Sect. 11.7.2 |
| \(\sigma_d\) | Population standard deviation of differences | Sect. 11.7.2 |
| \(R^2\) | R-squared | Sect. 16.4.2 |
| \(H_0\) | Null hypothesis | Sect. 28.2 |
| \(H_1\) | Alternative hypothesis | Sect. 28.2 |
| CI | Confidence interval | Chap. 24 |
| s.e. | Standard error | Def. 19.4 |
| \(n\) | Sample size | |
| \(\chi^2\) | The chi-squared test statistic | Sect. 31.6.3 |
| \(\pm\) | Plus-or-minus (give-or-take) | Sect. 22.3 |