記述統計：平均は合計を n で割ったもの、中央値は並べ替えたデータの真ん中の値です（n が偶数なら真ん中 2 つの平均）。標本標準偏差は s = √(Σ(x − x̄)²/(n − 1))。z スコア z = (x − μ)/σ は、値が平均から標準偏差何個分離れているかを表します。
標準化する
z = (77.5 − 70)/5 = 1.5
答え
z = 1.5
5.信頼区間
答え: E₂ = 3.21
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
誤差の幅の比
E₂/E₁ = (2.576/1.645)·√(64/256) = 0.783
新しい誤差の幅
E₂ = 4.1·0.783 = 3.21
答え
E₂ = 3.21
6.信頼区間
答え: 上限 = −0.2081 kg
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
プールした標準偏差
sp² = (5·4² + 19·2.5²)/24 = 8.281
sp = 2.878
誤差の幅
E = 1.711·2.878·√(1/6 + 1/20) = 2.292
区間
−2.5 ± 2.292 → [−4.792, −0.2081]
答え
上限 = −0.2081 kg
7.仮説検定と信頼区間
答え: z = −2.987
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.
差の標準誤差
SE = √(17²/40 + 17²/100) = √10.12 = 3.18
検定統計量
z = (41.5 − 51)/3.18 = −2.987
答え
z = −2.987
|z| > 1.96 なので、有意水準 5%（両側）で H₀ は棄却されます。
8.仮説検定と信頼区間
答え: z = −1.68
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.