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% 例題I5.1.12：散布図と相関関係（One More）★
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10名の社員に対して，2つの指標$\mathrm{X},\mathrm{Y}$でパフォーマンス評価を行ったところ，指標Xの評価値の分散は$\frac{47}{7}$，指標Yの評価値の分散は$\frac{47}{7}$で，XとYの評価値の共分散は$\frac{40}{7}$であった．このとき，XとYの評価値の相関係数を求めよ．また，XとYの評価値として対応する散布図を次の(i)〜(iii)から選べ．
(i)
(ii)
(iii)
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Xの評価値の分散が$\frac{47}{7}$より，標準偏差は，$\sqrt{\frac{47}{7}}$
Yの評価値の分散が$\frac{47}{7}$より，標準偏差は，$\sqrt{\frac{47}{7}}$
XとYの評価値の共分散が$\frac{40}{7}$より，求める相関係数は，
\[
\frac{\frac{40}{7}}{\sqrt{\frac{47}{7}}\times\sqrt{\frac{47}{7}}}=\frac{40}{47}
\]
また，相関係数は正で，正の相関があるため，散布図は(iii)
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% 問題I5.1.12：散布図と相関関係（One More）★
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あるクラスの10名の生徒について，2つの指標$\mathrm{X},\mathrm{Y}$でテスト結果を評価した．指標Xの評価値の分散は$\frac{33}{4}$，指標Yの評価値の分散は$\frac{33}{4}$で，XとYの評価値の共分散は$-\frac{15}{2}$であった．このとき，XとYの評価値の相関係数を求めよ．また，XとYの評価値として対応する散布図を次の(i)〜(iii)から選べ．
(i)
(ii)
(iii)
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% 問題I5.1.12の解答（One More）★
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Xの評価値の分散が$\frac{33}{4}$より，標準偏差は$\sqrt{\frac{33}{4}}$
Yの評価値の分散が$\frac{33}{4}$より，標準偏差は$\sqrt{\frac{33}{4}}$
XとYの評価値の共分散が$-\frac{15}{2}$より，求める相関係数は，
\[
\frac{-\frac{15}{2}}{\sqrt{\frac{33}{4}}\times\sqrt{\frac{33}{4}}}=-\frac{30}{33}=-\frac{10}{11}
\]
また，相関係数は負で，強い負の相関があるため，散布図は(i)
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