C.1 Symbols and standard errors
The following table lists the statistics used to estimate unknown population parameters.
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 | \(\bar{x}_1 - \bar{x}_2\) | \(\mu_1 - \mu_2\) | ✔ | \(\displaystyle \sqrt{\text{s.e.}(\bar{x}_1)^2 + \text{s.e.}(\bar{x}_2)^2}\) | Ch. 30 |
| Difference between proportions | \(\hat{p}_1 - \hat{p}_2\) | \(p_1 - p_2\) | ✔ | CI: \(\displaystyle \sqrt{\text{s.e.}(\hat{p}_1)^2 + \text{s.e.}(\hat{p}_2)^2}\) | Ch. 31 |
| ✔ | HT: \(\displaystyle \sqrt{\text{s.e.}(\hat{p}_1)^2 + \text{s.e.}(\hat{p}_2)^2}\) using common proportion \(\hat{p}\) | Ch. 31 | |||
| Odds ratio (OR) | 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 |