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 = (55 − 60)/5 = −1
答え
z = −1
5.信頼区間
答え: 上限 = 16.43 h
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.
点推定値
x̄₁ − x̄₂ = 61 − 50 = 11
標準誤差
SE = √(17²/100 + 20²/50) = 3.3
誤差の幅
E = 1.645·3.3 = 5.429
区間
11 ± 5.429 → [5.571, 16.43]
答え
上限 = 16.43 h
区間は 0 を含まないので、平均には有意な差があります。
6.信頼区間
答え: n = 722
Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.
n について解く
n = 1.645²·0.4·0.6/0.03² = 721.6
切り上げる
n = 722
答え
n = 722
7.仮説検定と信頼区間
答え: t = 1.615
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.