A confidence interval (CI) at level $(1-\alpha)\times100\%$ is a range of values computed from sample data, designed so that the procedure captures the true parameter in $(1-\alpha)\times100\%$ of repeated applications.
$$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
Critical values $z_{\alpha/2}$:
| Level | $\alpha$ | $z_{\alpha/2}$ |
|---|---|---|
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.960 |
| 99% | 0.01 | 2.576 |
Margin of error (half-width): $E = z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}$.
Example 1: Sample of $n=64$, $\bar{x} = 52.3$, $\sigma = 8$. Construct a 95% CI for $\mu$.
$\$52.3 \pm 1.96 \times \frac{8}{\sqrt{64}} = 52.3 \pm 1.96 \times 1 = 52.3 \pm 1.96 = (50.34,\ 54.26)$$
$$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
Example 2: $n = 200$, $x = 80$ successes. Construct a 95% CI for $p$.
$\hat{p} = 80/200 = 0.4$. Margin of error $= 1.96\sqrt{\dfrac{0.4\times0.6}{200}} = 1.96\times0.0346 \approx 0.068$.
95% CI: $(0.332,\ 0.468)$.
To achieve a margin of error $\leq E$:
$$n \geq \left(\frac{z_{\alpha/2}\cdot\sigma}{E}\right)^2$$
Example 3: For 95% CI on $\mu$ with $\sigma = 15$ and $E = 3$:
$$n \geq \left(\frac{1.96 \times 15}{3}\right)^2 = (9.8)^2 = 96.04 \Rightarrow n \geq 97$$
Correct interpretation: “We are 95% confident that the true population mean lies between $a$ and $b$.”
Incorrect interpretation: “There is a 95% probability that $\mu$ lies between $a$ and $b$.”
(The parameter $\mu$ is fixed; it either is or is not in the interval.)
Wider CI means:
- Higher confidence level (larger $z^*$), or
- Smaller sample size (larger $\sigma/\sqrt{n}$), or
- Larger population standard deviation
| Change | Effect on CI width |
|---|---|
| Increase $n$ | Narrower |
| Increase confidence level | Wider |
| Increase $\sigma$ | Wider |
KEY TAKEAWAY: A confidence interval gives a range of plausible values for an unknown parameter. The margin of error $E = z^* \cdot \sigma/\sqrt{n}$ decreases with larger sample size.
EXAM TIP: When constructing a CI, show the formula, substitute values clearly, and state the interval as a pair of numbers in the form $(a, b)$ or $a < \mu < b$.
COMMON MISTAKE: Interpreting 95% confidence as meaning “there is a 95% chance $\mu$ is in this specific interval.” The interval is random; $\mu$ is fixed. Confidence refers to the long-run frequency of the procedure.
