Carrying Out a Test for the Slope · 执行斜率检验
The test statistic
- The slope test statistic compares $b$ to $0$ in standard errors:
-
$$t = \frac{b}{SE_b}$$
- Both $b$ and $SE_b$ come straight from the computer output.
- Output often prints this $t$ (and its p-value) for you already.
检验统计量
- 斜率检验统计量以标准误为单位把 $b$ 与 $0$ 比较：
-
$$t = \frac{b}{SE_b}$$
- $b$ 和 $SE_b$ 都直接来自计算机输出。
- 输出常常已经为你打印了这个 $t$（和它的 p 值）。
Find the p-value
- Use the $t$-distribution with $df = n - 2$.
- Two-sided $H_a$: the standard output p-value (already two-tailed).
- One-sided $H_a$: halve the reported two-sided p-value (if $b$ is in the claimed direction).
- The p-value is the chance of a slope this far from $0$ if $\beta$ were truly $0$.
求 p 值
- 使用 $df = n - 2$ 的 $t$ 分布。
- **双侧 $H_a$：**标准输出的 p 值（已是双尾）。
- 单侧 $H_a$：把报告的双侧 p 值减半（如果 $b$ 在所声称的方向上）。
- p 值是：如果 $\beta$ 真的为 $0$，出现一个离 $0$ 这么远的斜率的机会。
Make a decision
- Compare the p-value to $\alpha$:
- p $\le \alpha$ → reject $H_0$: convincing evidence of a linear relationship.
- p $> \alpha$ → fail to reject: not convincing evidence.
- Standard decision rule.
做出决策
- 把 p 值与 $\alpha$ 比较：
- p $\le \alpha$ → 拒绝 $H_0$：有令人信服的线性关系证据。
- p $> \alpha$ → 不拒绝：证据不足。
- 标准决策规则。
Conclude in context
- State it about the linear relationship between the real variables.
- "Convincing evidence of a positive linear relationship between hours and score" (if rejected).
- Or "not convincing evidence of a linear relationship" (if not).
- Failing to reject never proves $\beta = 0$.
结合语境下结论
- 关于真实变量之间的线性关系来陈述。
- “有令人信服的证据表明小时数与分数之间存在正的线性关系”（若拒绝）。
- 或“没有令人信服的线性关系证据”（若不拒绝）。
- 不拒绝永远不能证明 $\beta = 0$。
Computer output's p-value is two-sided — halve it for a one-sided $H_a$ (and only if the sample slope points the way $H_a$ claims). Use $df = n - 2$, and read $t = b/SE_b$ from the output rather than recomputing. As always, "fail to reject" means insufficient evidence, not proof of no relationship.
计算机输出的 p 值是双侧的——对单侧 $H_a$ 要减半（且仅当样本斜率指向 $H_a$ 所声称的方向）。使用 $df = n - 2$，并从输出读 $t = b/SE_b$ 而非重新计算。一如既往，“不拒绝”意味着证据不足，而非没有关系的证明。
Output: $b = 4.2$, $SE_b = 1.0$; two-sided p-value $= 0.001$; $H_a: \beta > 0$, $\alpha = 0.05$.
- $t = \dfrac{4.2}{1.0} = 4.2$ ($df = n-2$).
- One-sided p-value: $0.001 / 2 = 0.0005$ ($b > 0$ matches $H_a$).
- Decide: $0.0005 \le 0.05$ → reject $H_0$; convincing evidence of a positive linear relationship.
输出：$b = 4.2$，$SE_b = 1.0$；双侧 p 值 $= 0.001$；$H_a: \beta > 0$，$\alpha = 0.05$。
- $t = \dfrac{4.2}{1.0} = 4.2$（$df = n-2$）。
- 单侧 p 值：$0.001 / 2 = 0.0005$（$b > 0$ 与 $H_a$ 一致）。
- 决策：$0.0005 \le 0.05$ → 拒绝 $H_0$；有令人信服的正线性关系证据。
Compute the slope $t = \frac{b}{SE_b}$ from the output, find the p-value at $df = n - 2$ (halve the two-sided output value for a one-sided $H_a$), and compare to $\alpha$: reject $H_0$ if p $\le \alpha$ (a linear relationship exists), else fail to reject — stated in context.
从输出计算斜率 $t = \frac{b}{SE_b}$，在 $df = n - 2$ 处求 p 值（对单侧 $H_a$ 把双侧输出值减半），并与 $\alpha$ 比较：p $\le \alpha$ 则拒绝 $H_0$（存在线性关系），否则不拒绝——结合语境陈述。
Testing the fitted slope against zero · 把拟合斜率与零作检验
t = b/SE_b measures how far the slope sits from 0. · t = b/SE_b 衡量斜率离 0 有多远。
b = 4.2, SE_b = 1.0. Compute the slope test statistic t = b/SE_b. · b = 4.2，SE_b = 1.0。计算斜率检验统计量 t = b/SE_b。
4.2 / 1.0 = 4.2. · 4.2 / 1.0 = 4.2。
A two-sided output p-value is 0.001. For a one-sided Ha (with b in the claimed direction), what is the p-value? · 双侧输出 p 值是 0.001。对单侧 Ha（b 在所声称方向上），p 值是多少？
Halve the two-sided value: 0.001 / 2 = 0.0005. · 把双侧值减半：0.001 / 2 = 0.0005。
With a p-value of 0.0005 and α = 0.05, the decision is... · p 值为 0.0005 且 α = 0.05，决策是……
0.0005 ≤ 0.05 → reject H0. · 0.0005 ≤ 0.05 → 拒绝 H0。
Computer regression output reports a two-sided p-value by default. · 计算机回归输出默认报告双侧 p 值。
Halve it for a one-sided alternative. · 对单侧备择假设把它减半。
Rejecting H0: β = 0 lets you conclude... · 拒绝 H0: β = 0 让你能得出……
Reject the flat-line null → a real linear relationship. · 拒绝平线零假设 → 一个真实的线性关系。