4.
![](https://www.ebk.net.cn/tikuimages/2/2018/500/shoutiniao28/ee7f1ef32e01b982fb66df226061d2e6.png)
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,得到如图所示的散点图及一些统计量的值.
\( \overline {x}\) | \( \overline {y}\) | \( \overline {w}\) | \( \sum\limits_{i=1}^{8}(x_{i}- \overline {x})^{2}\) | \( \sum\limits_{i=1}^{8}(w_{i}- \overline {w})^{2}\) | \( \sum\limits_{i=1}^{8}(x_{i}- \overline {x})(y_{1}- \overline {y})\) | \( \sum\limits_{i=1}^{8}(w_{i}- \overline {w})(y_{i}- \overline {y})\) |
\(46.6\) | \(563\) | \(6.8\) | \(289.8\) | \(1.6\) | \(1469\) | \(108.8\) |
其中\(w_{i}= \sqrt {x_{i}}\),\( \overline {w}= \dfrac {1}{8} \sum\limits_{i=1}^{8}w_{i}\)
\((\)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\()\)已知这种产品的年利润\(z\)与\(x\)、\(y\)的关系为\(z=0.2y-x.\)根据\((\)Ⅱ\()\)的结果回答下列问题,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\((u_{1},v_{1})\),\((u_{2},v_{2})\),\(…\),\((u_{n},v_{n})\),其回归直线\(v=α+βμ\)的斜率和截距的最小二乘估计分别为:\( ∧β= \dfrac { \sum\limits_{i=1}^{n}(u_{i}- \overline {u})(v_{i}- \overline {v})}{ \sum\limits_{i=1}^{n}(u_{i}- \overline {u})^{2}}\),\( ∧α= \overline {v}- ∧β \overline {u}\).