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.
\(\overrightarrow{x}\) | \(\overrightarrow{y}\) | \(\overrightarrow{w}\) | \(\sum\limits_{i=1}^{8}{{{({{x}_{i}}-\overline{x})}^{2}}}\) | \(\sum\limits_{i=1}^{8}{{{({{w}_{i}}-\overline{w})}^{2}}}\) | \(\sum\limits_{i=1}^{8}{({{x}_{i}}-\overline{x})({{y}_{i}}-\overline{y})}\) | \(\sum\limits_{i=1}^{8}{({{w}_{i}}-\overline{w})({{y}_{i}}-\overline{y})}\) |
\(46.6\) | \(56.3\) | \(6.8\) | \(289.8\) | \(1.6\) | \(1469\) | \(108.8\) |
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\()\)已知这种产品的年利率\(z\)与\(x\)、\(y\)的关系为\(z=0.2y-x.\)根据\((\)Ⅱ\()\)的结果回答下列问题:
\((ⅰ)\)年宣传费\(x=49\)时,年销售量及年利润的预报值是多少?
\((ⅱ)\)年宣传费\(x\)为何值时,年利率的预报值最大?
附:对于一组数据\(({{u}_{1}},{{v}_{1}})\),\(({{u}_{2}},{{v}_{2}})\),\(……\),\(({{u}_{n}},{{v}_{n}})\),其回归线\(v=\alpha +\beta u\)的斜率和截距的最小二乘估计分别为:\(\widehat{\beta }{=}\dfrac{\sum\limits_{i=1}^{n}{({{u}_{i}}-\overline{u})({{v}_{i}}-\overline{v)}}}{\sum\limits_{i=1}^{n}{{{({{u}_{i}}-\overline{u})}^{2}}}}\),\(\widehat{\alpha }{=}\overline{v}-\widehat{\beta }\overline{u}\)