某公司为确定下一年度投入某种产品的宣传费,需了解年宣传费\(x(\)单位:千元\()\)对年销售量\(y(\)单位:\(t)\)和年利润\(z(\)单位:千元\()\)的影响,对近\(8\)年的年宣传费\(x_{i}\)和年销售量\(y_{i}(i=1,2,…,8)\)数据作了初步处理,得到下面的散点图及一些统计量的值.
\( \overrightarrow{x}\) | \( \overrightarrow{y}\) | \( \overrightarrow{w}\) | \( \sum\limits_{i=1}^{8}\;(x_{1}- \overrightarrow{x})^{2}\) | \( \sum\limits_{i=1}^{8}\;(w_{1}- \overrightarrow{w})^{2}\) | \( \sum\limits_{i=1}^{8}\;(x_{1}- \overrightarrow{x})(y- \overrightarrow{y})\) | \( \sum\limits_{i=1}^{8}\;(w_{1}- \overrightarrow{w})(y- \overrightarrow{y})\) |
\(46.6\) | \(56.3\) | \(6.8\) | \(289.8\) | \(1.6\) | \(1469\) | \(108.8\) |
表中\(w_{i}= \sqrt {x_{i}}\),\( \overrightarrow{w}= \dfrac {1}{8} \sum\limits_{i=1}^{8}w_{i}\)
\((\)Ⅰ\()\)根据散点图判断,\(y=a+bx\)与\(y=c+d \sqrt {x}\)哪一个适宜作为年销售量\(y\)关于年宣传费\(x\)的回归方程类型?\((\)给出判断即可,不必说明理由\()\)
\((\)Ⅱ\()\)根据\((\)Ⅰ\()\)的判断结果及表中数据,建立\(y\)关于\(x\)的回归方程;
\((\)Ⅲ\()\)以知这种产品的年利率\(z\)与\(x\)、\(y\)的关系为\(z=0.2y-x.\)根据\((\)Ⅱ\()\)的结果回答
当年宣传费\(x=49\)时,年销售量及年利润的预报值是多少?
附:对于一组数据\((u_{1}\) \(v_{1})\),\((u_{2}\) \(v_{2})…..(u_{n}\) \(v_{n})\),其回归线\(v=α+βu\)的斜率和截距的最小二乘估计分别为:\( \hat β= \dfrac { \sum\limits_{i=1}^{n}(u_{i}- \overset{ .}{u})(v_{i}- \overset{ .}{v})}{ \sum\limits_{i=1}^{n}(u_{i}- \overset{}{u})^{2}}, \hat α= \overset{ .}{v}- \hat β \overset{ .}{u}\).