10.
某工厂生产不同规格的一种产品,根据检测标准,其合格产品的质量\(y(g)\)与尺寸\(x(mm)\)之间近似满足关系式\(y=a{{x}^{b}}(a,b\)为大于\(0\)的常数\()\),现随机抽取\(6\)件合格产品,测得数据如下:
尺寸\(( \) \(mm\) \()\) | \(38\) | \(48\) | \(58\) | \(68\) | \(78\) | \(88\) |
质量\(( \) \(g\) \()\) | \(16.8\) | \(18.8\) | \(20.7\) | \(22.4\) | \(24.0\) | \(25.5\) |
对数据作了初步处理,相关统计量的值如下表:
\(\sum\limits_{i=1}^{6}{(\ln {{x}_{i}}\ln {{y}_{i}})}\) | \(\sum\limits_{i=1}^{6}{(\ln {{x}_{i}})}\) | \(\sum\limits_{i=1}^{6}{(\ln {{y}_{i}})}\) | \(\sum\limits_{i=1}^{6}{{{(\ln {{x}_{i}})}^{2}}}\) |
\(75.3\) | \(24.6\) | \(18.3\) | \(101.4\) |
\((\)Ⅰ\()\)根据所给数据,求\(y\)关于\(x\)的回归方程;
\((\)Ⅱ\()\)按照某项指标测定,所抽取的\(6\)件合格品中有\(3\)件是优等品,现从这\(6\)件合格品中任取\(3\)件,记\(X\)为取到优等品的件数,求随机变量\(X\)的分布列和数学期望.
附:对于一组数据\(({v}_{1},{u}_{1}),({v}_{2},{u}_{2}),⋯,({v}_{n},{u}_{n}) \),其回归直线\(u=\alpha +\beta v\)的斜率和截距的最小二乘估计值分别为\(\hat{\beta }=\dfrac{\sum\limits_{i=1}^{n}{{{v}_{i}}{{u}_{i}}}-n\bar{v}\cdot \bar{u}}{\sum\limits_{i=1}^{n}{v_{i}^{2}}-n{{{\bar{v}}}^{2}}}\),\(\hat{\alpha }=\bar{u}-\hat{\beta }\bar{v}\).