收录高三寒假作业20新题
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20240125-103720
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20240125-104329
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"id": "023772",
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"content": "判断正误(正确的画``\\checkmark'', 错误的画``$\\times$'')\\\\\n(1) 相关关系与函数关系都是一种确定性的关系, 也是一种因果关系;\\blank{20}\\\\\n(2) 利用散点图可以直观判断两个变量的关系是否可以用线性关系表示;\\blank{20}\\\\\n(3) 只有两个变量有相关关系, 所得到的回归模型才有预测价值;\\blank{20}\\\\\n(4) 事件 $X$、$Y$ 的关系越密切, 由观测数据计算得到的 $\\chi^2$ 的观测值越大;\\blank{20}\\\\\n(5) 通过回归方程 $y=\\hat{a}x+\\hat{b}$ 可以估计和观测变量的取值和变化趋势.\\blank{20}",
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"content": "观察下列各图形, 其中两个变量 $x$、$y$ 具有相关关系的图是\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.31/1,0.56/0.84,0.71/0.55,1.09/0.24,1.48/0.31,1.26/0.41,1.37/0.64,1.07/0.53,0.94/0.71,1.15/0.76,1.02/0.92,1.28/0.85,1.4/1.02,1.28/1.16}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{1}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.28/0.68,0.46/1.01,0.49/0.6,0.41/0.31,0.7/0.46,0.65/0.82,0.93/1.01,0.9/0.66,0.81/0.35,1/0.4,1.11/0.57}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{2}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.2/0.4,0.45/0.51,0.65/0.67,0.78/0.9,0.91/0.86,0.96/1.08,1.09/1.13,1.21/1.27}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{3}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.17/1.44,0.24/1.3,0.36/1.2,0.5/1.09,0.64/0.96,0.82/0.84,0.88/0.68,0.99/0.55,1.14/0.45}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}",
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"023774": {
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"id": "023774",
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"content": "某公司在 $2020$ 年上半年的月收入 $x$ (单位:万元)与月支出 $y$ (单位:万元)的统计资料如表所示, 根据统计资料, 则\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 月份 & 1 月 & 2 月 & 3月 & 4 月 & 5 月 & 6 月 \\\\\n\\hline 月收入 $x$ & 12.3 & 14.5 & 15.0 & 17.0 & 19.8 & 20.6 \\\\\n\\hline 月支出 $y$ & 5.63 & 5.75 & 5.82 & 5.89 & 6.11 & 6.18 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\onech{月收入的中位数是 $15, x$ 与 $y$ 有正线性相关关系}{月收入的中位数是 $17, x$ 与 $y$ 有负线性相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有正线性相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有负线性相关关系}",
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"genre": "选择题",
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"023775": {
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"id": "023775",
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"content": "某研究机构对高三学生的记忆力 $x$ 和判断力 $y$ 进行统计分析, 所得数据如表, 则 $y$ 对 $x$ 的线性回归直线方程为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$ & 6 & 8 & 10 & 12 \\\\\n\\hline$y$ & 2 & 3 & 5 & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{center}",
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"023776": {
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"id": "023776",
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"content": "某沙漠地区经过治理, 生态系统得到很大改善, 野生动物数量有所增加. 为调查该地区某种野生动物的数量, 将其分成面积相近的 $200$ 个地块, 从这些地块中用简单随机抽样的方法抽取 $20$ 个作为样区, 调查得到样本数据 $(x_i, y_i)$($i=1$、$2$、$\\cdots, 20$), 其中 $x_i$ 和 $y_i$ 分别表示第 $i$ 个样区的植物覆盖面积(单位:公顷)和这种野生动物的数量, 并计算得\n$\\displaystyle\\sum_{i=1}^{20}x_i=60$, $\\displaystyle\\sum_{i=1}^{20}y_i=1200$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline{x})^2=80$, $\\displaystyle\\sum_{i=1}^{20}(y_i-\\overline{y})^2=9000$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline{x}) \\cdot(y_i-\\overline{y})=800$.\\\\\n(1) 求该地区这种野生动物数量的估计值(这种野生动物数量的估计值等于样区这种野生动物数量的平均数乘以地块数);\\\\\n(2) 求样本 $(x_i, y_i)$($i=1$、$2$、$\\cdots$、$20$) 的相关系数 (精确到 $0.01$).\\\\\n附: 相关系数\n$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})(y_i-\\overline{y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})^2 \\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})^2}}$.",
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"023777": {
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"id": "023777",
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"content": "有人收集了某 $10$ 年中某城市居民年收人(即该城市所有居民在一年内收人的总和)与某种商品的销售额的相关数据如表, \n且已知 $\\displaystyle\\sum_{i=1}^{10}x_i=380.0$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 第$n$年 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 年收人 $x$ / 亿元 & 32.0 & 31.0 & 33.0 & 36.0 & 37.0 & 38.0 & 39.0 & 43.0 & 45.0 & $x_{10}$ \\\\\n\\hline 商品销售额 $y /$ 万元 & 25.0 & 30.0 & 34.0 & 37.0 & 39.0 & 41.0 & 42.0 & 44.0 & 48.0 & $y_{10}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 求第 $10$ 年的年收人 $x_{10}$;\\\\\n(2) 若该城市居民年收人 $x$ 与该种商品的销售额 $y$ 之间满足线性回归方程 $y=\\dfrac{363}{254}x+\\hat{b}$,\\\\\n\\textcircled{1} 求该种商品第 $10$ 年的销售额 $y_{10}$;\\\\\n\\textcircled{2} 若该城市居民年收人为 $40.0$ 亿元, 估计这种商品的销售额是多少?(精确到 $0.01$ 万元).\\\\\n附: (i) 在线性回归方程 $y=\\hat{a}x+\\hat{b}$ 中, $\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n x_i y_i-n \\overline{x}\\cdot \\overline{y}}{\\displaystyle\\sum^n x^2}$, $\\hat{b}=\\hat{y}-\\hat{a}\\overline{x}$, $\\displaystyle\\sum_{i=1}^n x_i^2-n \\overline{x}^2$;\\\\\n(ii) $\\displaystyle\\sum_{i=1}^{10}x_i^2-10 x^{-2}=254.0$, $\\displaystyle\\sum_{i=1}^9 x_i y_i=12875.0$, $\\displaystyle\\sum_{i=1}^9 y_i=340.0$.",
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"023778": {
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"id": "023778",
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"content": "某学生兴趣小组随机调查了某市 $100$ 天中每天的空气质量等级和当天到某公园锻炼的人次, 整理数据得到下表(单位:天):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline 空气质量等级 &{$[0,200]$}& $(200,400]$ & $(400,600]$ \\\\\n\\hline 1 (优) & 2 & 16 & 25 \\\\\n\\hline 2 (良) & 5 & 10 & 12 \\\\\n\\hline 3 (轻度污染) & 6 & 7 & 8 \\\\\n\\hline 4 (中度污染) & 7 & 2 & 0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 分别估计该市一天的空气质量等级为 $1,2,3,4$ 的概率;\\\\\n(2) 求一天中到该公园锻炼的平均人次的估计值(同一组中的数据用该组区间的中点值为代表);\\\\\n(3) 若某天的空气质量等级为 1 或 2 , 则称这天``空气质量好'';若某天的空气质量等级为 3 或 4 , 则称这天``空气质量不好''. 根据所给数据, 完成下面的 $2 \\times 2$ 列联表, 并根据列联表, 判断是否有 $95 \\%$ 的把握认为一天中到该公园锻炼的人次与该市当天的空气质量有关?\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\\hline & 人次 $\\leq 400$ & 人次 $>400$ \\\\\n\\hline 空气质量好 & & \\\\\n\\hline 空气质量不好 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{tabular}{|l|l|l|l|}\\hline$P$($\\chi^2 \\geq k$) & $0.050$ & $0.010$ & $0.001$ \\\\\n\\hline $k$ & $3.841$ & $6.635$ & $10.828$ \\\\\n\\hline\n\\end{tabular}",
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"030001": {
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"id": "030001",
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"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
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