{
  "api_version": "v1",
  "publisher": {
    "name": "Manazoo",
    "url": "https://manazoo.com/"
  },
  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "quiz",
  "unit": {
    "slug": "statistical-inference",
    "title": "統計的な推測",
    "url": "https://manazoo.com/high/math-b/statistical-inference/"
  },
  "canonical_url": "https://manazoo.com/high/math-b/statistical-inference/quiz/",
  "questions": [
    {
      "id": "q-bd-mean",
      "type": "input",
      "prompt": "B(4,1/2)に従う確率変数Xの平均を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "平均はnp＝4×1/2＝2。",
      "entities": [
        "binomial-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-bd-p-x2",
      "type": "input",
      "prompt": "4枚の硬貨を投げ、表の出る枚数をXとする。P(X＝2)を求めよ。",
      "choices": null,
      "answer": "3/8",
      "accept": [],
      "explanation": "P(X＝2)＝₄C₂(1/2)²(1/2)²＝6×1/16＝3/8。",
      "entities": [
        "binomial-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-bd-sum-of-bernoulli",
      "type": "choice",
      "prompt": "二項分布B(n,p)に従う確率変数Xは、何の和とみなせるか。",
      "choices": [
        "母平均の推定値",
        "n個の正規分布の和",
        "n個の独立なベルヌーイ型確率変数の和",
        "1個の確率変数をn倍した値"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "1回ごとの試行を0か1の確率変数とみて、独立にn個足したものが二項分布になる。",
      "entities": [
        "binomial-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-bd-variance",
      "type": "input",
      "prompt": "B(4,1/2)に従う確率変数Xの分散を求めよ。",
      "choices": null,
      "answer": "1",
      "accept": [],
      "explanation": "分散はnpq＝4×1/2×1/2＝1。",
      "entities": [
        "binomial-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cipm-half-width",
      "type": "input",
      "prompt": "σ/√n＝4のとき、信頼度95%の信頼区間の半分の幅（1.96×σ/√n）を求めよ。",
      "choices": null,
      "answer": "7.84",
      "accept": [],
      "explanation": "1.96×4＝7.84。",
      "entities": [
        "confidence-interval-estimation-of-population-mean"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cipm-lower-bound",
      "type": "input",
      "prompt": "標本平均108、半分の幅7.84のとき、95%信頼区間の下限を求めよ。",
      "choices": null,
      "answer": "100.16",
      "accept": [],
      "explanation": "下限は108−7.84＝100.16。",
      "entities": [
        "confidence-interval-estimation-of-population-mean"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cipm-n-up-interval-narrow",
      "type": "choice",
      "prompt": "標本の大きさnを大きくすると、信頼区間の幅はどうなるか。",
      "choices": [
        "母標準偏差により決まり、nとは無関係",
        "狭くなる",
        "広くなる",
        "変わらない"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "幅は1.96×σ/√nの2倍。nが大きくなるほどσ/√nが小さくなるので、幅は狭くなる。",
      "entities": [
        "confidence-interval-estimation-of-population-mean"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cipm-upper-bound",
      "type": "input",
      "prompt": "標本平均108、半分の幅7.84のとき、95%信頼区間の上限を求めよ。",
      "choices": null,
      "answer": "115.84",
      "accept": [],
      "explanation": "上限は108+7.84＝115.84。",
      "entities": [
        "confidence-interval-estimation-of-population-mean"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-htnd-judge-1pct",
      "type": "choice",
      "prompt": "有意水準を1%に変えると、確率0.014との比較結果はどうなるか。",
      "choices": [
        "変わらない",
        "判断できない",
        "0.014は1%(0.01)より大きいので帰無仮説を否定できない",
        "0.014は1%より小さいので否定できる"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "0.014は1%（0.01）より大きいので、この基準では帰無仮説を否定できない。有意水準で判断は変わる。",
      "entities": [
        "hypothesis-testing-with-normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-htnd-judge-5pct",
      "type": "choice",
      "prompt": "有意水準5%のとき、確率0.014から帰無仮説の判断はどれか。",
      "choices": [
        "判断できない",
        "標本が足りない",
        "帰無仮説を否定できる（対立仮説が正しいと考える）",
        "帰無仮説を否定できない"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "0.014は有意水準5%（0.05）より小さいので、帰無仮説を否定し、対立仮説が正しいと考える。",
      "entities": [
        "hypothesis-testing-with-normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-htnd-p-value",
      "type": "input",
      "prompt": "P(0≤Z≤2.2)＝0.486のとき、P(Z≥2.2)を求めよ。",
      "choices": null,
      "answer": "0.014",
      "accept": [],
      "explanation": "P(Z≥2.2)＝0.5−0.486＝0.014。",
      "entities": [
        "hypothesis-testing-with-normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-htnd-z-value",
      "type": "input",
      "prompt": "n＝100,p＝1/2の二項分布でX＝61のとき、Z＝(X−50)/5の値を求めよ。",
      "choices": null,
      "answer": "2.2",
      "accept": [],
      "explanation": "Z＝(61−50)/5＝11/5＝2.2。",
      "entities": [
        "hypothesis-testing-with-normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-nd-p-between",
      "type": "input",
      "prompt": "P(0≤Z≤1.96)＝0.475のとき、P(−1.96≤Z≤1.96)を求めよ。",
      "choices": null,
      "answer": "0.95",
      "accept": [],
      "explanation": "対称性より、両側の確率を足して0.475×2＝0.95。",
      "entities": [
        "normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-nd-p-z-le-196",
      "type": "input",
      "prompt": "P(0≤Z≤1.96)＝0.475のとき、P(Z≤1.96)を求めよ。",
      "choices": null,
      "answer": "0.975",
      "accept": [],
      "explanation": "P(Z≤0)＝0.5なので、P(Z≤1.96)＝0.5＋0.475＝0.975。",
      "entities": [
        "normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-nd-sigma-rule",
      "type": "choice",
      "prompt": "確率変数Xが正規分布N(μ,σ²)に従うとき、P(μ−σ≤X≤μ+σ)の近似値はどれか。",
      "choices": [
        "0.997",
        "0.5",
        "0.683",
        "0.954"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "μ±σの範囲に入る確率はおよそ0.683。μ±2σなら0.954、μ±3σなら0.997になる。",
      "entities": [
        "normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-nd-standardize",
      "type": "input",
      "prompt": "確率変数Xが正規分布N(50,25²)に従うとき、X＝99に対応するZの値を求めよ。",
      "choices": null,
      "answer": "1.96",
      "accept": [],
      "explanation": "Z＝(X−μ)/σ＝(99−50)/25＝49/25＝1.96。",
      "entities": [
        "normal-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-psmd-n-up-sd-down",
      "type": "choice",
      "prompt": "標本の大きさnを大きくすると、標本平均の標準偏差はどうなるか。",
      "choices": [
        "変わらない",
        "小さくなる",
        "0になる",
        "大きくなる"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "標本平均の標準偏差はσ/√n。nが大きくなるほど√nが大きくなるので、全体の値は小さくなる。",
      "entities": [
        "population-and-sample-mean-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-psmd-sd-n100",
      "type": "input",
      "prompt": "母標準偏差σ＝20、標本の大きさn＝100のとき、標本平均の標準偏差を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "標本平均の標準偏差はσ/√n＝20/√100＝20/10＝2。",
      "entities": [
        "population-and-sample-mean-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-psmd-sd-of-mean",
      "type": "input",
      "prompt": "母標準偏差σ＝20、標本の大きさn＝25のとき、標本平均の標準偏差を求めよ。",
      "choices": null,
      "answer": "4",
      "accept": [],
      "explanation": "標本平均の標準偏差はσ/√n＝20/√25＝20/5＝4。",
      "entities": [
        "population-and-sample-mean-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-psmd-standardize",
      "type": "input",
      "prompt": "母平均100、標本平均の標準偏差4のとき、標本平均が108に対応するZの値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "Z＝(X̄−m)/(σ/√n)＝(108−100)/4＝2。",
      "entities": [
        "population-and-sample-mean-distribution"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvev-e-x-squared",
      "type": "input",
      "prompt": "3枚の硬貨の例で、E(X²)を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "E(X²)＝0²×1/8＋1²×3/8＋2²×3/8＋3²×1/8＝0＋3/8＋12/8＋9/8＝3。",
      "entities": [
        "random-variable-expectation-and-variance"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvev-expected-value",
      "type": "input",
      "prompt": "3枚の硬貨の例で、期待値E(X)を求めよ。",
      "choices": null,
      "answer": "3/2",
      "accept": [],
      "explanation": "E(X)＝0×1/8＋1×3/8＋2×3/8＋3×1/8＝12/8＝3/2。",
      "entities": [
        "random-variable-expectation-and-variance"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvev-two-formulas-relation",
      "type": "choice",
      "prompt": "分散を求める2つの式、Σ(xk−m)²pkとE(X²)−{E(X)}²の関係はどれか。",
      "choices": [
        "E(X²)の式の方が必ず大きい",
        "常に同じ値になる",
        "標本が大きいときだけ同じ",
        "独立なときだけ同じ値になる"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "(xk−m)²を展開して整理すると、2つの式は代数的に常に等しいことが分かる。独立性や標本の大きさとは関係ない。",
      "entities": [
        "random-variable-expectation-and-variance"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvev-variance-direct",
      "type": "input",
      "prompt": "3枚の硬貨の例で、分散V(X)を偏差の2乗の式で求めよ。",
      "choices": null,
      "answer": "3/4",
      "accept": [],
      "explanation": "それぞれの偏差を2乗して確率をかけて足すと、9/32＋3/32＋3/32＋9/32＝24/32＝3/4。",
      "entities": [
        "random-variable-expectation-and-variance"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvpd-multiple-of-three",
      "type": "choice",
      "prompt": "1個のさいころを投げ、出た目が3の倍数なら1、そうでなければ0とする確率変数をYとする。P(Y＝1)はどれか。",
      "choices": [
        "1/2",
        "1/3",
        "1/6",
        "2/3"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "1から6のうち3の倍数は3と6の2通り。6通り中2通りなので2/6＝1/3。",
      "entities": [
        "random-variables-and-probability-distributions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvpd-p-x1",
      "type": "input",
      "prompt": "2枚の硬貨を投げ、表の出る枚数をXとする。P(X＝1)を求めよ。",
      "choices": null,
      "answer": "1/2",
      "accept": [],
      "explanation": "表が1枚になる出方は（表,裏）（裏,表）の2通り。4通り中2通りなので2/4＝1/2。",
      "entities": [
        "random-variables-and-probability-distributions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvpd-p0-plus-p2",
      "type": "input",
      "prompt": "2枚の硬貨の例で、P(X＝0)＋P(X＝2)を求めよ。",
      "choices": null,
      "answer": "1/2",
      "accept": [],
      "explanation": "確率分布はP(X＝0)＝1/4、P(X＝2)＝1/4なので、足すと1/4＋1/4＝1/2。",
      "entities": [
        "random-variables-and-probability-distributions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-rvpd-sum-one",
      "type": "choice",
      "prompt": "確率変数Xの確率分布で、必ず成り立つ性質はどれか。",
      "choices": [
        "ΣP(X＝xk)は1になる",
        "P(X＝xk)は全て等しい",
        "ΣP(X＝xk)は0になる",
        "P(X＝xk)はxkに比例する"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "確率分布の確率を全部足すと、必ず起こる事象の確率として1になる。比例や値がすべて等しいとは限らない。",
      "entities": [
        "random-variables-and-probability-distributions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-sirv-e-of-sum",
      "type": "input",
      "prompt": "E(X)＝1、E(W)＝1/2のとき、E(X+W)を求めよ。",
      "choices": null,
      "answer": "3/2",
      "accept": [],
      "explanation": "確率変数の和の期待値は、独立かどうかに関わらずE(X+W)＝E(X)+E(W)＝1+1/2＝3/2。",
      "entities": [
        "sum-and-independence-of-random-variables"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-sirv-e-of-y",
      "type": "input",
      "prompt": "確率変数Xについて E(X)＝1 のとき、Y＝2X+3の期待値E(Y)を求めよ。",
      "choices": null,
      "answer": "5",
      "accept": [],
      "explanation": "E(Y)＝E(2X+3)＝2E(X)+3＝2×1+3＝5。",
      "entities": [
        "sum-and-independence-of-random-variables"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-sirv-independence-def",
      "type": "choice",
      "prompt": "確率変数X,Yが独立であるとは何か。",
      "choices": [
        "E(X)＝E(Y)であること",
        "Xの値がYの値より常に大きいこと",
        "V(X)＝V(Y)であること",
        "P(X＝a,Y＝b)＝P(X＝a)P(Y＝b)が常に成り立つこと"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "独立の定義は、どんなa,bの組についても確率が積になることだ。期待値や分散が等しいかとは関係ない。",
      "entities": [
        "sum-and-independence-of-random-variables"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-sirv-v-of-y",
      "type": "input",
      "prompt": "確率変数Xについて V(X)＝1/2 のとき、Y＝2X+3の分散V(Y)を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "V(Y)＝V(2X+3)＝2²V(X)＝4×1/2＝2。bを足しても分散は変わらない。",
      "entities": [
        "sum-and-independence-of-random-variables"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}