A Confidence Interval for p
| English | Chinese | Pinyin |
|---|---|---|
| confidence interval | 置信区间 | zhì xìn qū jiān |
| standard error | 标准误 | biāo zhǔn wù |
| margin of error | 误差幅度 | wù chā fú dù |
| confidence level | 置信水平 | zhì xìn shuǐ píng |
Check the conditions first
- Before a confidence interval 置信区间 for $p$, verify three conditions:
- Random: the data come from a random sample (or randomized experiment).
- 10%: if sampling without replacement, $n \le 0.10N$.
- Large counts: $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$.
Estimate and standard error
- The point estimate is the sample proportion $\hat{p}$.
- The standard error 标准误 (estimated SD of $\hat{p}$) is:
-
$$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
- We use $\hat{p}$ here because the true $p$ is unknown.
Build the interval
- A confidence interval has the form estimate $\pm$ margin of error:
-
$$\hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
- $z^{*}$ is the critical value for your confidence level ($1.96$ for $95\%$).
- The result is a range of plausible values for $p$.
Margin of error and level
- The margin of error 误差幅度 is $z^{*} \times SE$ — the "$\pm$" half-width.
- The confidence level 置信水平 (e.g. $95\%$) sets $z^{*}$ and how often the method works.
- Higher confidence → larger $z^{*}$ → wider interval.
- Larger $n$ → smaller $SE$ → narrower interval.
A confidence interval is estimate $\pm$ margin of error, not $\pm$ the standard error alone. You must multiply the $SE$ by the critical value $z^{*}$. Using $z^{*}=1.96$ for $95\%$ is standard; forgetting it (or using the wrong level's $z^{*}$) makes the interval the wrong width and the stated confidence a lie.
A sample of $n = 100$ gives $\hat{p} = 0.60$; build a $95\%$ interval.
- SE: $\sqrt{\dfrac{0.6 \times 0.4}{100}} = \sqrt{0.0024} \approx 0.049$.
- Margin of error: $1.96 \times 0.049 \approx 0.096$.
- Interval: $0.60 \pm 0.096 = (0.504,\ 0.696)$.
After checking random, 10%, and large counts, a one-sample $z$-interval for $p$ is $\hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. The margin of error is $z^{*} \times SE$; a higher confidence level widens the interval, and a larger sample narrows it.
The central confidence area
A 95% interval captures the middle 95% via z* = 1.96.
n = 100, p̂ = 0.60. Compute the standard error √(p̂(1−p̂)/n). Two decimals.
√(0.6·0.4/100) = √0.0024 ≈ 0.049 ≈ 0.05.
With SE ≈ 0.049 and z* = 1.96, find the margin of error z*·SE. Two decimals.
1.96 × 0.049 ≈ 0.096 ≈ 0.10.
A confidence interval has the form...
You must multiply SE by the critical value z*.
Raising the confidence level (say 95% → 99%) makes the interval...
Higher confidence → larger z* → wider interval.
For a 95% confidence interval, what critical value z* do you use? (two decimals)
z* = 1.96 for 95% confidence.