Setting Up a Test for the Slope · 为斜率设立检验
The slope hypotheses
- A test about a slope asks whether there's a real linear relationship.
- Null $H_0: \beta = 0$ — the slope is zero.
- Alternative $H_a$: $\beta \ne 0$ (some relationship), or one-sided $\beta > 0$ / $\beta < 0$.
- The default $H_0$ is almost always "$\beta = 0$."
斜率假设
- 关于斜率的检验问的是是否存在真实的线性关系。
- 零假设 $H_0: \beta = 0$——斜率为零。
- 备择假设 $H_a$：$\beta \ne 0$（存在某种关系），或单侧的 $\beta > 0$ / $\beta < 0$。
- 默认的 $H_0$ 几乎总是“$\beta = 0$”。
Why β = 0 means "no relationship"
- If the true slope $\beta = 0$, the regression line is flat.
- A flat line predicts the same $y$ no matter what $x$ is → $x$ tells you nothing.
- So $\beta = 0$ is exactly "no linear relationship."
- Rejecting $H_0$ means the data support a real linear link.
为什么 β = 0 意味着“没有关系”
- 如果真实斜率 $\beta = 0$，回归线是平的。
- 一条平线无论 $x$ 是什么都预测相同的 $y$ → $x$ 什么也没告诉你。
- 所以 $\beta = 0$ 恰好是“没有线性关系”。
- 拒绝 $H_0$ 意味着数据支持一个真实的线性联系。
Verify the conditions
- Check the same LINER conditions as for the interval:
- Linear, Independent, Normal, Equal SD, Random.
- Use residual plots for the linear / normal / equal-SD parts.
- Only then is the $t$-model for the slope valid.
验证条件
- 检查与区间相同的 LINER 条件：
- Linear（线性）、Independent（独立）、Normal（正态）、Equal SD（等标准差）、Random（随机）。
- 用残差图检查线性 / 正态 / 等标准差部分。
- 只有这样，斜率的 $t$ 模型才有效。
The right test
- The procedure is a $t$-test for the slope, with $df = n - 2$.
- It's the regression counterpart of the one-mean $t$-test.
- The test statistic (next lesson) compares $b$ to $0$ in units of $SE_b$.
- Everything is read from the same computer output.
正确的检验
- 程序是斜率的 $t$ 检验，$df = n - 2$。
- 它是单均值 $t$ 检验在回归中的对应。
- 检验统计量（下一课）以 $SE_b$ 为单位把 $b$ 与 $0$ 比较。
- 一切都从同一份计算机输出中读取。
The null is $\beta = 0$ (a flat line = no linear relationship), and the $df$ is $n - 2$. Don't test "$b = 0$" — $b$ is the sample estimate; the hypothesis is about the population slope $\beta$. And choose a one- or two-sided $H_a$ from the question before seeing the output.
**零假设是 $\beta = 0$（平线 = 没有线性关系），$df$ 是 $n - 2$。**不要检验“$b = 0$”——$b$ 是样本估计；假设是关于总体斜率 $\beta$ 的。并在看到输出之前就从问题中选定单侧或双侧的 $H_a$。
Does studying more relate to higher scores?
- $H_0: \beta = 0$ (no linear relationship between hours and score).
- $H_a: \beta > 0$ (more hours → higher scores) — a one-sided claim.
- Conditions checked, we'll run a $t$-test for the slope with $df = n - 2$.
学得越多和分数越高有关吗？
- $H_0: \beta = 0$（小时数与分数之间没有线性关系）。
- $H_a: \beta > 0$（更多小时 → 更高分数）——一个单侧主张。
- 条件检查后，我们将用 $df = n - 2$ 做斜率的 $t$ 检验。
A slope test states $H_0: \beta = 0$ (no linear relationship) vs. a one- or two-sided $H_a$, using a $t$-test for the slope with $df = n - 2$ after checking the LINER conditions. A slope of $0$ means a flat line — $x$ gives no help predicting $y$.
斜率检验陈述 $H_0: \beta = 0$（没有线性关系）对单侧或双侧的 $H_a$，在检查 LINER 条件后，使用 $df = n - 2$ 的斜率 $t$ 检验。斜率为 $0$ 意味着平线——$x$ 无助于预测 $y$。
The null: a flat line (β = 0) · 零假设：一条平线（β = 0）
H0 says the true slope is 0 — no linear relationship. · H0 说真实斜率是 0——没有线性关系。
The usual null hypothesis for a slope test is... · 斜率检验通常的零假设是……
H0 is about the population slope β being 0. · H0 是关于总体斜率 β 为 0。
A true slope β = 0 means... · 真实斜率 β = 0 意味着……
Flat line → x doesn't help predict y. · 平线 → x 无助于预测 y。
The slope t-test uses what degrees of freedom for n = 20? · 斜率 t 检验对 n = 20 用多少自由度？
df = n − 2 = 18. · df = n − 2 = 18。
The hypotheses are about the population slope β, not the sample slope b. · 假设是关于总体斜率 β，而非样本斜率 b。
b is the estimate; β is the parameter tested. · b 是估计；β 是被检验的参数。
Rejecting H0: β = 0 provides evidence that... · 拒绝 H0: β = 0 提供了……的证据。
Rejecting the flat-line null supports a real linear link. · 拒绝平线零假设支持一个真实的线性联系。