Difference of Two Proportions · 两个比例之差
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| difference of two sample proportions/ˈdɪfrəns ɒv tuː ˈsæmpl prəˈpɔːʃnz/ | 两个样本比例之差 | liǎng gè yàng běn bǐ lì zhī chà |
Comparing two groups
- Often we compare two proportions — say, support in group $1$ vs group $2$.
- The statistic is the difference of two sample proportions 两个样本比例之差 $\hat{p}_1 - \hat{p}_2$.
- Its sampling distribution is centered at the true difference: $\mu = p_1 - p_2$.
- It lets us judge whether two groups genuinely differ.
比较两个组
- 我们常常比较两个比例——比如组 $1$ 与组 $2$ 的支持率。
- 统计量是两个样本比例之差 $\hat{p}_1 - \hat{p}_2$。
- 它的抽样分布以真实差为中心：$\mu = p_1 - p_2$。
- 它让我们判断两个组是否真的不同。
Combining the spreads
- The standard deviation combines both samples' variability:
-
$$\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$$
- Note the $+$: variances add (echoing the "combining variables" rule) even for a difference.
- Each group contributes its own $\frac{p(1-p)}{n}$ term.
合并两个分散
- 标准差把两个样本的变异性合并起来：
-
$$\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$$
- 注意那个 $+$：即使是差，方差也相加（呼应“组合变量”法则）。
- 每个组贡献它自己的 $\frac{p(1-p)}{n}$ 项。
Check both samples
- Apply the large counts condition to each group: $n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2)$ all $\ge 10$.
- Require independence: the two samples are independent of each other.
- Also the 10% condition for each if sampling without replacement.
- Only then is the normal model for $\hat{p}_1 - \hat{p}_2$ valid.
检查两个样本
- 对每个组应用大计数条件：$n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2)$ 都 $\ge 10$。
- 要求独立性：两个样本彼此独立。
- 不放回抽样时每个组还要满足 10% 条件。
- 只有这样，$\hat{p}_1 - \hat{p}_2$ 的正态模型才有效。
Using the normal model
- With conditions met, $\hat{p}_1 - \hat{p}_2$ is approximately normal.
- Standardize with a $z$-score: (observed difference $-$ true difference) $\div$ the combined SD.
- Then read probabilities from the standard normal, as before.
- This is the foundation for two-proportion tests and intervals (Unit 6).
使用正态模型
- 条件满足时，$\hat{p}_1 - \hat{p}_2$ 近似正态。
- 用 $z$ 分数标准化：（观察到的差 $-$ 真实的差）$\div$ 合并的标准差。
- 然后像之前一样从标准正态读出概率。
- 这是两比例检验和区间（第 6 单元）的基础。
Variances add — even for a difference. The SD of $\hat{p}_1-\hat{p}_2$ combines the two $\frac{p(1-p)}{n}$ terms with a $+$, then a square root; you never subtract them. And check the large counts condition in both groups separately (four counts, all $\ge 10$) — one healthy group doesn't excuse a thin one.
方差相加——即使是差。$\hat{p}_1-\hat{p}_2$ 的标准差用 $+$ 把两个 $\frac{p(1-p)}{n}$ 项合并，再开平方根；你绝不相减它们。并且要在两个组里分别检查大计数条件（四个计数，都 $\ge 10$）——一个健康的组不能替一个人数少的组开脱。
$p_1 = 0.5$ ($n_1 = 100$), $p_2 = 0.4$ ($n_2 = 100$).
- Center: $p_1 - p_2 = 0.5 - 0.4 = 0.1$.
- SD: $\sqrt{\dfrac{0.5(0.5)}{100} + \dfrac{0.4(0.6)}{100}} = \sqrt{0.0025 + 0.0024} = \sqrt{0.0049} = 0.07$.
- Counts: all four ($50, 50, 40, 60$) exceed $10$ → normal is OK.
$p_1 = 0.5$（$n_1 = 100$），$p_2 = 0.4$（$n_2 = 100$）。
- 中心：$p_1 - p_2 = 0.5 - 0.4 = 0.1$。
- 标准差：$\sqrt{\dfrac{0.5(0.5)}{100} + \dfrac{0.4(0.6)}{100}} = \sqrt{0.0025 + 0.0024} = \sqrt{0.0049} = 0.07$。
- **计数：**四个（$50, 50, 40, 60$）都超过 $10$ → 可用正态。
The difference of two sample proportions $\hat{p}_1 - \hat{p}_2$ has mean $p_1 - p_2$ and SD $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$ (variances add). Model it as normal when the large counts and independence conditions hold for both samples.
两个样本比例之差 $\hat{p}_1 - \hat{p}_2$ 的均值为 $p_1 - p_2$，标准差为 $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$（方差相加）。当大计数和独立性条件对两个样本都成立时，把它建模为正态。
Distribution of p-hat-1 − p-hat-2 · p-hat-1 − p-hat-2 的分布
Centered at p1 − p2; the two variances add under the square root. · 以 p1 − p2 为中心；两个方差在平方根下相加。
p1=0.5, p2=0.4. Find the center (mean) of p-hat-1 − p-hat-2. · p1=0.5，p2=0.4。求 p-hat-1 − p-hat-2 的中心（均值）。
Mean = p1 − p2 = 0.5 − 0.4 = 0.1. · 均值 = p1 − p2 = 0.5 − 0.4 = 0.1。
With p1=0.5,n1=100 and p2=0.4,n2=100: SD = √(0.0025+0.0024). Round to two decimals. · p1=0.5,n1=100 且 p2=0.4,n2=100：标准差 = √(0.0025+0.0024)。保留两位小数。
√0.0049 = 0.07. · √0.0049 = 0.07。
To get the SD of a difference of two proportions, you subtract the two variances. · 求两个比例之差的标准差时，你要相减两个方差。
Variances ADD, even for a difference, then square-root. · 方差相加，即使是差，再开平方根。
For a normal model of p-hat-1 − p-hat-2, the large counts condition must hold... · 对 p-hat-1 − p-hat-2 的正态模型，大计数条件必须……
All four counts (both groups) must be ≥ 10. · 四个计数（两个组）都必须 ≥ 10。
The two samples must be independent of each other for this model. · 这个模型要求两个样本彼此独立。
Independence between samples is required. · 样本之间需要独立。