このページは、数研出版：数学Ｂ[710]
第２章 統計的な推測
第２章 統計的な推測
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数研出版数学Ｂ 第１章 数列
数研出版数学Ｂ 第２章 統計的な推測
第２章 統計的な推測
第１節 確率分布
p.55 練習1\(\begin{array}{c|cccc|c}
X & 110 & 100 & 10 & 0 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & 1
\end{array}\)
解法のPoint｜確率変数と確率分布
X & 110 & 100 & 10 & 0 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & \displaystyle\frac{\,1\,}{\,4\,} & 1
\end{array}\)
解法のPoint｜確率変数と確率分布
p.59 練習4\(~V(X)=\displaystyle \frac{\,9\,}{\,25\,}~,~\sigma (X)=\displaystyle \frac{\,3\,}{\,5\,}\)
■ この問題の詳しい解説はこちら！
■ この問題の詳しい解説はこちら！
p.60 練習5\(~V(X)=\displaystyle \frac{\,9\,}{\,25\,}~,~\sigma (X)=\displaystyle \frac{\,3\,}{\,5\,}\)
■ この問題の詳しい解説はこちら！
■ この問題の詳しい解説はこちら！
p.60 深める標準偏差 \(\sigma (X)\) の値が小さいと、分散 \(V(X)\) の値も小さくなる
また、分散の定義より、確率変数と期待値との差 \(\{x_k-m\}\) が小さくなるので確率変数 \(X\) のとる値は分布の平均の近くに集中する
また、分散の定義より、確率変数と期待値との差 \(\{x_k-m\}\) が小さくなるので確率変数 \(X\) のとる値は分布の平均の近くに集中する
p.62 練習6\({\small (1)}~E(Y)=\displaystyle \frac{\,9\,}{\,2\,}~,~V(Y)=\displaystyle \frac{\,35\,}{\,12\,}~,~\)
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,6\,}\)
\({\small (2)}~E(Y)=-7~,~V(Y)=\displaystyle \frac{\,35\,}{\,3\,}\)
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,3\,}~,~\)
\({\small (3)}~E(Y)=\displaystyle \frac{\,17\,}{\,2\,}~,~V(Y)=\displaystyle \frac{\,105\,}{\,4\,}~,~\)
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,2\,}\)
解法のPoint｜確率変数の変換
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,6\,}\)
\({\small (2)}~E(Y)=-7~,~V(Y)=\displaystyle \frac{\,35\,}{\,3\,}\)
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,3\,}~,~\)
\({\small (3)}~E(Y)=\displaystyle \frac{\,17\,}{\,2\,}~,~V(Y)=\displaystyle \frac{\,105\,}{\,4\,}~,~\)
\(\sigma(Y)=\displaystyle \frac{\,\sqrt{105}\,}{\,2\,}\)
解法のPoint｜確率変数の変換
p.64 練習7\(~~~~~\begin{array}{c|ccc|c}~~~~~~~Y~\\[-5pt] ~X~~~ & ~~1~~ & ~~2~~ & ~~3~~ & ~~計~~ \\[5pt]
\hline
1 & 0 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,6\,} & 0 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,1\,}{\,6\,} & 0 & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & 1
\end{array}\)
解法のPoint｜2つの確率変数の同時分布
\hline
1 & 0 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,6\,} & 0 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,1\,}{\,6\,} & 0 & \displaystyle\frac{\,2\,}{\,6\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & 1
\end{array}\)
解法のPoint｜2つの確率変数の同時分布
p.67 問3\(a=1~,~b=1\) のときは例5より成り立つ
\(a=0~,~2\) のとき、\(~P(X=a)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(b=0~,~2\) のとき、\(~P(X=b)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)
また、\(a=1\) と \(b=0~,~2\) の任意の組合せで、\(~P(X=a)=\displaystyle \frac{\,8\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,2\,}\)\(~P(Y=b)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(~P(X=a~,~Y=b)=\displaystyle \frac{\,2\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,8\,}\)\(a=0~,~2\) と \(b=1\) の任意の組合せで、\(~P(X=a)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(~P(Y=b)=\displaystyle \frac{\,8\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,2\,}\)\(~P(X=a~,~Y=b)=\displaystyle \frac{\,2\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,8\,}\)
したがって、\(X\) のとる任意の値 \(a\) と \(Y\) のとる任意の値 \(b\) について、\(~P(X=a~,~Y=b)=P(X=a)P(Y=b)\)が成り立つ
\(a=0~,~2\) のとき、\(~P(X=a)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(b=0~,~2\) のとき、\(~P(X=b)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)
また、\(a=1\) と \(b=0~,~2\) の任意の組合せで、\(~P(X=a)=\displaystyle \frac{\,8\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,2\,}\)\(~P(Y=b)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(~P(X=a~,~Y=b)=\displaystyle \frac{\,2\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,8\,}\)\(a=0~,~2\) と \(b=1\) の任意の組合せで、\(~P(X=a)=\displaystyle \frac{\,4\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,4\,}\)\(~P(Y=b)=\displaystyle \frac{\,8\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,2\,}\)\(~P(X=a~,~Y=b)=\displaystyle \frac{\,2\,}{\,16\,}=\displaystyle \frac{\,1\,}{\,8\,}\)
したがって、\(X\) のとる任意の値 \(a\) と \(Y\) のとる任意の値 \(b\) について、\(~P(X=a~,~Y=b)=P(X=a)P(Y=b)\)が成り立つ
p.66 練習11\(X~,~Y\) の同時分布は、次のようになる
解法のPoint｜独立な確率変数
\(\begin{array}{c|cccccc|c}~~~~~~~Y~\\[-5pt] ~X~~~ & ~~1~~ & ~~2~~ & ~~3~~ & ~~4~~ & ~~5~~ & ~~6~~ & ~~計~~ \\[5pt]
\hline
1 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
4 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
5 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
6 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & 1
\end{array}\)
\hline
1 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
4 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
5 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
6 & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & \displaystyle\frac{\,6\,}{\,36\,} & 1
\end{array}\)
これより、\(a=1\) 〜 \(6\) と \(b=1\) 〜 \(6\) の任意の組合せについて、\(~P(X=a)=\displaystyle \frac{\,1\,}{\,6\,}~,~P(Y=b)=\displaystyle \frac{\,1\,}{\,6\,}\)\(~P(X=a~,~Y=b)=\displaystyle \frac{\,1\,}{\,36\,}\)であるから、\(~P(X=a~,~Y=b)=P(X=a)P(Y=b)\)が成り立つ
解法のPoint｜独立な確率変数
p.71 練習12\(~P(B)=\displaystyle \frac{\,1\,}{\,6\,}~,~P(C)=\displaystyle \frac{\,5\,}{\,36\,}\)
\(~P(B\cap C)=\displaystyle \frac{\,1\,}{\,36\,}\)
これより、２つの事象 \(B\) と \(C\) は従属である
解法のPoint｜事象の独立・従属
\(~P(B\cap C)=\displaystyle \frac{\,1\,}{\,36\,}\)
これより、２つの事象 \(B\) と \(C\) は従属である
解法のPoint｜事象の独立・従属
p.77 練習18\({\small (1)}~E(X)=3~,~V(X)=\displaystyle \frac{\,9\,}{\,4\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,3\,}{\,2\,}\)
\({\small (2)}~E(X)=\displaystyle \frac{\,9\,}{\,2\,}~,~V(X)=\displaystyle \frac{\,9\,}{\,4\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,3\,}{\,2\,}\)
\({\small (3)}~E(X)=\displaystyle \frac{\,16\,}{\,3\,}~,~V(X)=\displaystyle \frac{\,16\,}{\,9\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,4\,}{\,3\,}\)
解法のPoint｜二項分布の期待値(平均)と分散・標準偏差
\(\sigma(X)=\displaystyle \frac{\,3\,}{\,2\,}\)
\({\small (2)}~E(X)=\displaystyle \frac{\,9\,}{\,2\,}~,~V(X)=\displaystyle \frac{\,9\,}{\,4\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,3\,}{\,2\,}\)
\({\small (3)}~E(X)=\displaystyle \frac{\,16\,}{\,3\,}~,~V(X)=\displaystyle \frac{\,16\,}{\,9\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,4\,}{\,3\,}\)
解法のPoint｜二項分布の期待値(平均)と分散・標準偏差
p.77 練習19\({\small (1)}~B\left(10~,~\displaystyle \frac{\,1\,}{\,2\,}\right)~,~E(X)=5~,~\)
\(\sigma(X)=\displaystyle \frac{\,\sqrt{10}\,}{\,2\,}\)
\({\small (2)}~B\left(50~,~\displaystyle \frac{\,3\,}{\,100\,}\right)~,~E(X)=\displaystyle \frac{\,3\,}{\,2\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,\sqrt{582}\,}{\,20\,}\)
解法のPoint｜硬貨を複数回投げる二項分布
\(\sigma(X)=\displaystyle \frac{\,\sqrt{10}\,}{\,2\,}\)
\({\small (2)}~B\left(50~,~\displaystyle \frac{\,3\,}{\,100\,}\right)~,~E(X)=\displaystyle \frac{\,3\,}{\,2\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,\sqrt{582}\,}{\,20\,}\)
解法のPoint｜硬貨を複数回投げる二項分布
p.80 練習20\({\small (1)}~P(0≦X≦1)=0.2~,~\)
\(P(1≦X≦3)=0.4\)
\({\small (2)}~P(0≦X≦1)=\displaystyle \frac{\,1\,}{\,4\,}~,~\)
\(P(1≦X≦2)=\displaystyle \frac{\,3\,}{\,4\,}\)
解法のPoint｜確率密度関数と確率
\(P(1≦X≦3)=0.4\)
\({\small (2)}~P(0≦X≦1)=\displaystyle \frac{\,1\,}{\,4\,}~,~\)
\(P(1≦X≦2)=\displaystyle \frac{\,3\,}{\,4\,}\)
解法のPoint｜確率密度関数と確率
p.80 問4\(~E(X)=\displaystyle \frac{\,2\,}{\,3\,}~,~V(X)=\displaystyle \frac{\,1\,}{\,18\,}\)\(\sigma(X)=\displaystyle \frac{\,\sqrt{2}\,}{\,6\,}\)
解法のPoint｜確率密度関数と期待値(平均)・分散
解法のPoint｜確率密度関数と期待値(平均)・分散
p.80 練習21\({\small (1)}~E(X)=\displaystyle \frac{\,5\,}{\,2\,}~,~V(X)=\displaystyle \frac{\,25\,}{\,12\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,5\sqrt{3}\,}{\,6\,}\)
\({\small (2)}~E(X)=\displaystyle \frac{\,4\,}{\,3\,}~,~V(X)=\displaystyle \frac{\,2\,}{\,9\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,\sqrt{2}\,}{\,3\,}\)
解法のPoint｜確率密度関数と期待値(平均)・分散
\(\sigma(X)=\displaystyle \frac{\,5\sqrt{3}\,}{\,6\,}\)
\({\small (2)}~E(X)=\displaystyle \frac{\,4\,}{\,3\,}~,~V(X)=\displaystyle \frac{\,2\,}{\,9\,}~,~\)
\(\sigma(X)=\displaystyle \frac{\,\sqrt{2}\,}{\,3\,}\)
解法のPoint｜確率密度関数と期待値(平均)・分散
p.83 練習22\({\small (1)}~0.8413\) \({\small (2)}~0.3085\)
\({\small (3)}~0.1359\) \({\small (4)}~0.6826\)
\({\small (5)}~0.9544\) \({\small (6)}~0.9973\)
解法のPoint｜標準正規分布と確率
\({\small (3)}~0.1359\) \({\small (4)}~0.6826\)
\({\small (5)}~0.9544\) \({\small (6)}~0.9973\)
解法のPoint｜標準正規分布と確率
p.83 練習23\({\small (1)}~0.9544\) \({\small (2)}~0.6554\)
\({\small (3)}~0.3811\)
解法のPoint｜正規分布と標準正規分布
\({\small (3)}~0.3811\)
解法のPoint｜正規分布と標準正規分布
p.84 練習24\({\small (1)}~\)約 \(9\) % \({\small (2)}~\)約 \(42\) %
\({\small (3)}~\)\(148~{\rm cm}\) 以下
解法のPoint｜正規分布の確率を求める文章問題
\({\small (3)}~\)\(148~{\rm cm}\) 以下
解法のPoint｜正規分布の確率を求める文章問題
問題
p.87 問題 1 期待値 \(\displaystyle \frac{\,35\,}{\,18\,}\)、分散 \(\displaystyle \frac{\,665\,}{\,324\,}\)
■ この問題の詳しい解説はこちら！
■ この問題の詳しい解説はこちら！
p.87 問題 2 期待値 \(\displaystyle \frac{\,12\,}{\,5\,}\)、分散 \(\displaystyle \frac{\,3\,}{\,5\,}\)
解法のPoint｜確率変数の和の期待値(平均)
解法のPoint｜独立な確率変数の和の分散・標準偏差
解法のPoint｜確率変数の和の期待値(平均)
解法のPoint｜独立な確率変数の和の分散・標準偏差
p.87 問題 3\({\small (1)}~\)期待値 \(58\)、分散 \(144\)
\({\small (2)}~\)期待値 \(-20\)、分散 \(16\)
\({\small (3)}~\)期待値 \(0\)、分散 \(1\)
解法のPoint｜確率変数の変換
解法のPoint｜二項分布の期待値(平均)と分散・標準偏差
\({\small (2)}~\)期待値 \(-20\)、分散 \(16\)
\({\small (3)}~\)期待値 \(0\)、分散 \(1\)
解法のPoint｜確率変数の変換
解法のPoint｜二項分布の期待値(平均)と分散・標準偏差
第２節 統計的な推測
p.90 練習26\(\begin{array}{c|cccc|c}
X & 1 & 2 & 3 & 4 & 計 \\
\hline
P & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,3\,}{\,10\,} & \displaystyle\frac{\,4\,}{\,10\,} & \displaystyle\frac{\,1\,}{\,10\,} & 1
\end{array}\)
母平均は \({\displaystyle \frac{\,12\,}{\,5\,}}\)、母標準偏差は \({\displaystyle \frac{\,\sqrt{21}\,}{\,5\,}}\)
解法のPoint｜母集団分布と母平均・母標準偏差
X & 1 & 2 & 3 & 4 & 計 \\
\hline
P & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,3\,}{\,10\,} & \displaystyle\frac{\,4\,}{\,10\,} & \displaystyle\frac{\,1\,}{\,10\,} & 1
\end{array}\)
母平均は \({\displaystyle \frac{\,12\,}{\,5\,}}\)、母標準偏差は \({\displaystyle \frac{\,\sqrt{21}\,}{\,5\,}}\)
解法のPoint｜母集団分布と母平均・母標準偏差
p.91 問6\({\small (1)}~\)
\(\begin{array}{c|cccc|c}~~~~~~~X_2~\\[-5pt] ~X_1~~~ & ~~1~~ & ~~2~~ & ~~3~~ & ~~4~~ & ~~計~~ \\[5pt]
\hline
1 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
4 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & 1
\end{array}\)
解法のPoint｜復元抽出と非復元抽出
\(\begin{array}{c|cccc|c}~~~~~~~X_2~\\[-5pt] ~X_1~~~ & ~~1~~ & ~~2~~ & ~~3~~ & ~~4~~ & ~~計~~ \\[5pt]
\hline
1 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
4 & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & 1
\end{array}\)
\({\small (2)}~\)
\(\begin{array}{c|cccc|c}~~~~~~~X_2~\\[-5pt] ~X_1~~~ & ~~1~~ & ~~2~~ & ~~3~~ & ~~4~~ & ~~計~~ \\[5pt]
\hline
1 & 0 & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} \\[5pt]
2 & \displaystyle\frac{\,1\,}{\,12\,} & 0 & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} \\[5pt]
3 & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & 0 & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} \\[5pt]
4 & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & \displaystyle\frac{\,1\,}{\,12\,} & 0 & \displaystyle\frac{\,3\,}{\,12\,} \\[5pt]
\hline
計 & \displaystyle\frac{\,3\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} & \displaystyle\frac{\,3\,}{\,12\,} & 1
\end{array}\)
解法のPoint｜復元抽出と非復元抽出
p.93 問7\({\small (1)}~\)
解法のPoint｜標本平均の確率分布
\(\begin{array}{c|ccccccc|c}
\overline{X} & 1 & \displaystyle\frac{\,3\,}{\,2\,} & 2 & \displaystyle\frac{\,5\,}{\,2\,} & 3 & \displaystyle\frac{\,7\,}{\,2\,} & 4 & 計 \\[5pt]
\hline
P & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,2\,}{\,16\,} & \displaystyle\frac{\,3\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,3\,}{\,16\,} & \displaystyle\frac{\,2\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & 1
\end{array}\)
\overline{X} & 1 & \displaystyle\frac{\,3\,}{\,2\,} & 2 & \displaystyle\frac{\,5\,}{\,2\,} & 3 & \displaystyle\frac{\,7\,}{\,2\,} & 4 & 計 \\[5pt]
\hline
P & \displaystyle\frac{\,1\,}{\,16\,} & \displaystyle\frac{\,2\,}{\,16\,} & \displaystyle\frac{\,3\,}{\,16\,} & \displaystyle\frac{\,4\,}{\,16\,} & \displaystyle\frac{\,3\,}{\,16\,} & \displaystyle\frac{\,2\,}{\,16\,} & \displaystyle\frac{\,1\,}{\,16\,} & 1
\end{array}\)
\({\small (2)}~\)
\(\begin{array}{c|ccccc|c}
\overline{X} & \displaystyle\frac{\,3\,}{\,2\,} & 2 & \displaystyle\frac{\,5\,}{\,2\,} & 3 & \displaystyle\frac{\,7\,}{\,2\,} & 計 \\[5pt]
\hline
P & \displaystyle\frac{\,2\,}{\,12\,} & \displaystyle\frac{\,2\,}{\,12\,} & \displaystyle\frac{\,4\,}{\,12\,} & \displaystyle\frac{\,2\,}{\,12\,} & \displaystyle\frac{\,2\,}{\,12\,} & 1
\end{array}\)
解法のPoint｜標本平均の確率分布
p.95 練習28 期待値 \({\displaystyle \frac{\,1\,}{\,10\,}}\)、標準偏差 \({\displaystyle \frac{\,3\,}{\,10\sqrt{n}\,}}\)
解法のPoint｜標本平均の文章問題
解法のPoint｜標本平均の文章問題
問題
p.109 問題 7\({\small (1)}~\)
\(\begin{array}{c|ccc|c}
X & 1 & 2 & 3 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,3\,}{\,6\,} & 1
\end{array}\)
\({\small (2)}~\)
母平均 \(\displaystyle \frac{\,7\,}{\,3\,}\)、母標準偏差 \(\displaystyle \frac{\,\sqrt{5}\,}{\,3\,}\)
解法のPoint｜母集団分布と母平均・母標準偏差
\(\begin{array}{c|ccc|c}
X & 1 & 2 & 3 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,6\,} & \displaystyle\frac{\,2\,}{\,6\,} & \displaystyle\frac{\,3\,}{\,6\,} & 1
\end{array}\)
\({\small (2)}~\)
母平均 \(\displaystyle \frac{\,7\,}{\,3\,}\)、母標準偏差 \(\displaystyle \frac{\,\sqrt{5}\,}{\,3\,}\)
解法のPoint｜母集団分布と母平均・母標準偏差
p.109 問題 8\({\small (1)}~m=\displaystyle \frac{\,74\,}{\,5\,}~,~\sigma=\displaystyle \frac{\,2\sqrt{26}\,}{\,5\,}\)
\({\small (2)}~\)
\(\begin{array}{c|ccccc|c}
\overline{X} & 13 & 14 & 15 & 16 & 17 & 計 \\
\hline
P & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,3\,}{\,10\,} & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,1\,}{\,10\,} & 1
\end{array}\)
\({\small (3)}~\)
\(E(\overline{X})=\displaystyle \frac{\,74\,}{\,5\,}~,~\sigma(\overline{X})=\displaystyle \frac{\,\sqrt{39}\,}{\,5\,}\)
解法のPoint｜標本平均の期待値(平均)と標準偏差
\({\small (2)}~\)
\(\begin{array}{c|ccccc|c}
\overline{X} & 13 & 14 & 15 & 16 & 17 & 計 \\
\hline
P & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,3\,}{\,10\,} & \displaystyle\frac{\,2\,}{\,10\,} & \displaystyle\frac{\,1\,}{\,10\,} & 1
\end{array}\)
\({\small (3)}~\)
\(E(\overline{X})=\displaystyle \frac{\,74\,}{\,5\,}~,~\sigma(\overline{X})=\displaystyle \frac{\,\sqrt{39}\,}{\,5\,}\)
解法のPoint｜標本平均の期待値(平均)と標準偏差
演習問題 統計的な推測
p.110 演習問題A 4 期待値 \(\displaystyle \frac{\,17\,}{\,10\,}\)、標準偏差 \(\displaystyle \frac{\,\sqrt{41}\,}{\,20\,}\)
解法のPoint｜標本平均の確率分布
解法のPoint｜標本平均の確率分布
p.111 演習問題B 7\({\small (1)}~\)
■ この問題の詳しい解説はこちら！
\(\begin{array}{c|cccccc|c}
X & 1 & 2 & 3 & 4 & 5 & 6 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,3\,}{\,36\,} & \displaystyle\frac{\,5\,}{\,36\,} & \displaystyle\frac{\,7\,}{\,36\,} & \displaystyle\frac{\,9\,}{\,36\,} & \displaystyle\frac{\,11\,}{\,36\,} & 1
\end{array}\)
X & 1 & 2 & 3 & 4 & 5 & 6 & 計 \\
\hline
P & \displaystyle\frac{\,1\,}{\,36\,} & \displaystyle\frac{\,3\,}{\,36\,} & \displaystyle\frac{\,5\,}{\,36\,} & \displaystyle\frac{\,7\,}{\,36\,} & \displaystyle\frac{\,9\,}{\,36\,} & \displaystyle\frac{\,11\,}{\,36\,} & 1
\end{array}\)
\({\small (2)}~\)期待値 \(\displaystyle \frac{\,161\,}{\,36\,}\)、標準偏差 \(\displaystyle \frac{\,\sqrt{2555}\,}{\,36\,}\)
■ この問題の詳しい解説はこちら！
p.111 演習問題B 11\({\small (1)}~[593~,~823]\) ただし、単位は票
\({\small (2)}~\)Aの支持率の方が高いと判断できる
■ この問題の詳しい解説はこちら！
\({\small (2)}~\)Aの支持率の方が高いと判断できる
■ この問題の詳しい解説はこちら！
