为了对\(2016\)年某校中考成绩进行分析,在\(60\)分以上的全体同学中随机抽出\(8\)位,他们的数学分数\((\)已折算为百分制\()\)从小到大排是\(60\)、\(65\)、\(70\)、\(75\)、\(80\)、\(85\)、\(90\)、\(95\),物理分数从小到大排是\(72\)、\(77\)、\(80\)、\(84\)、\(88\)、\(90\)、\(93\)、\(95\).
\((1)\)若规定\(85\)分以上为优秀,求这\(8\)位同学中恰有\(3\)位同学的数学和物理分数均为优秀的概率;
\((2)\)若这\(8\)位同学的数学、物理、化学分数事实上对应如下表:
学生编号 | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
数学分数\(x\) | \(60\) | \(65\) | \(70\) | \(75\) | \(80\) | \(85\) | \(90\) | \(95\) |
物理分数\(y\) | \(72\) | \(77\) | \(80\) | \(84\) | \(88\) | \(90\) | \(93\) | \(95\) |
化学分数\(z\) | \(67\) | \(72\) | \(76\) | \(80\) | \(84\) | \(87\) | \(90\) | \(92\) |
\(①\)用变量\(y\)与\(x\)、\(z\)与\(x\)的相关系数说明物理与数学、化学与数学的相关程度;
\(②\)求\(y\)与\(x\)、\(z\)与\(x\)的线性回归方程\((\)系数精确到\(0.01)\),当某同学的数学成绩为\(50\)分时,估计其物理、化学两科的得分.
参考公式:相关系数\(r= \dfrac { \sum\limits_{i=1}^{n}(x_{i}- \overset{}{x})(y_{i}- \overset{}{y})}{ \sqrt { \sum\limits_{i=1}^{n}(x_{i}- \overset{}{x})^{2}}\cdot \sum\limits_{i=1}^{n}(y_{i}- \overset{}{y})^{2}}\),
回归直线方程是:\( \overset{\hat{} }{y}=bx+a\),其中\(b= \dfrac { \sum\limits_{i=1}^{n}(x_{i}- \overset{}{x})(y_{i}- \overset{}{y})}{ \sum\limits_{i=1}^{n}(x_{i}- \overset{}{x})^{2}},a= \overset{ .}{y}-b \overset{ .}{x}\),
参考数据:\( \overset{ .}{x}=77.5, \overset{ .}{y}=85, \overset{ .}{z}=81, \sum\limits_{i=1}^{8}(x_{i}- \overset{ .}{x})^{2}≈1050, \sum\limits_{i=1}^{8}(y_{i}- \overset{ .}{y})^{2}≈456\),\( \sum\limits_{i=1}^{8}(z_{i}- \overset{ .}{z})^{2}≈550, \sum\limits_{i=1}^{8}(x_{i}- \overset{ .}{x})(y_{i}- \overset{ .}{y})≈688\),\( \sum\limits_{i=1}^{8}(x_{i}- \overset{ .}{x})(z_{i}- \overset{ .}{z})≈755, \sqrt {1050}≈32.4\),\( \sqrt {456}≈21.4, \sqrt {550}≈23.5\).