解题方法
1 . 为了研究某种细菌随天数
变化的繁殖个数
,收集数据如下:
(1)在图中作出繁殖个数
关于天数
变化的散点图,并由散点图判断
(
为常数)与
(
为常数,且
)哪一个适宜作为繁殖个数
关于天数
变化的回归方程类型?(给出判断即可,不必说明理由)
(2)对于非线性回归方程
(
为常数,且
),令
,可以得到繁殖个数的对数z关于天数x具有线性关系及一些统计量的值.
(ⅰ)证明:“对于非线性 回归方程
,令
,可以得到繁殖个数的对数
关于天数
具有线性 关系(即
为常数)”;
(ⅱ)根据(ⅰ)的判断结果及表中数据,建立
关于
的回归方程(系数保留2位小数).
附:对于一组数据
,其回归直线方程
的斜率和截距的最小二乘估计分别为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
天数![]() | 1 | 2 | 3 | 4 | 5 | 6 |
繁殖个数![]() | 6 | 12 | 25 | 49 | 95 | 190 |
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/7/4bacce99-5592-4ee8-bcc4-a60b71687351.png?resizew=149)
(1)在图中作出繁殖个数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce3a7d89965c094792f87594bd68afd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bda9896c593ee0a75ac2db7ddd6ee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31091823724baad3b893de6214ed243d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0de04adc20200028d604f7bf08582aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2de2538d41d5fbc30744fc8c0e4fe3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)对于非线性回归方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31091823724baad3b893de6214ed243d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0de04adc20200028d604f7bf08582aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2de2538d41d5fbc30744fc8c0e4fe3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54d28162b2a8309f0f7f193e733be414.png)
![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
3.50 | 62.83 | 3.53 | 17.50 | 596.57 | 12.09 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31091823724baad3b893de6214ed243d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54d28162b2a8309f0f7f193e733be414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b73c7432ba7e04d20bda1363ffa48bc.png)
(ⅱ)根据(ⅰ)的判断结果及表中数据,建立
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
附:对于一组数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69f22ea836f2025901725da985790579.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95523e41adf5e135049d4097a07f189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c5a0bfe22905260f047609a4180c21e.png)
您最近一年使用:0次
2 . 2023年高考进入倒计时,为了帮助学子们在紧张的备考中放松身心,某重点高中通过开展形式多样的减压游戏,确保同学们以稳定心态,良好地状态迎战高考,游戏规则如下:盒子中初始装有2个白球和1个红球各一个,每次有放回的任取一个,连续取两次,将以上过程记为一轮.如果每一轮取到的两个球都是红球,则记该轮为成功,否则记为失败.在抽取过程中,如果某一轮成功,则停止;否则,在盒子中再放入一个白球,然后接着进行下一轮抽球,如此不断继续下去,直至成功.
(1)如果某同学进行该抽球游戏时,最多进行三轮,即使第三轮不成功,也停止抽球,记其进行抽球试验的轮次数为随机变量
,求
的分布列和数学期望;
(2)为验证抽球试验成功的概率不超过
,假设有1000名学生独立的进行该抽球试验,记
表示成功时抽球试验的轮次数,
表示对应的人数,部分统计数据如下:
求
关于
的回归方程
,并通过回归方程预测成功的总人数(
取整数部分);
(3)证明:
.
附:经验回归方程系数:
,
;
参考数据:
,
,
(其中
,
).
(1)如果某同学进行该抽球游戏时,最多进行三轮,即使第三轮不成功,也停止抽球,记其进行抽球试验的轮次数为随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)为验证抽球试验成功的概率不超过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![]() | 1 | 2 | 3 | 4 | 5 |
![]() | 120 | 62 | 33 | 20 | 15 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd5d2b05c1bb3d089b0b44ee0f1309b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3617671ab9daae844ca0a46066fe7a.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd5dedcf1b362cc68eacc6e886eb9465.png)
附:经验回归方程系数:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a3f1e4697dce21f74266debbdc70c57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1db6103cb0f1d2bd6b19235d53ee7e98.png)
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd25720b00bf1598feecc6b3eef7fe8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e22c63f3a796ae8d1cfa9e2769b26be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9141a6d3fd0084de0bfa9136e826865a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9dd2642b6808b94f8f09d6726ef3e0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0593a949e9ca8680f71bad6725806bd.png)
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