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.
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 $0$.
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.
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 $\beta = 0$.
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.
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.
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.
推定された傾きが0と比較される
t = b/SE_b は、傾きが0からどれだけ離れているかを測る。
b = 4.2, SE_b = 1.0。傾き検定統計量 t = b/SE_b を計算せよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
4.2 / 1.0 = 4.2.
両側出力のp値が0.001の場合。片側のHa（bが主張された方向にある場合）におけるp値はいくらか？
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
両側の値を半分に割る：0.001 / 2 = 0.0005。
p値が0.0005、α = 0.05の場合、判断は...
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
0.0005 ≤ 0.05 → H0を棄却する。
コンピュータによる回帰出力では、デフォルトで両側のp値が報告される。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
片側検定の場合にはこれを半分にする。
H0: β = 0 を棄却すれば…と結論づけられる。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
水平な帰無仮説を棄却する → 真の線形関係である。