![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f14db37344529d273e36d835241d0d39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca98d80a6b806a27c46ae078dd6dabe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
A:两球的颜色相同且号码相邻;B: 两球的颜色相同,但号码不相邻;
C: 两球的颜色不同,但号码相邻;D: 两球的号码相同;E: 其它情况.
经营者打算将以上五种类别中最不容易发生的一种类别对应一等奖,最容易发生的一种类别对应二等奖,其他类别对应三等奖.
(1)一、二等奖分别对应哪一种类别(用字母表示即可);
(2)若一、二、三等奖分别获得价值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0efba7147f5b9ced8bc4a72f0a9fb8af.png)
2 . 某纺织厂为了生产一种高端布料,准备从A农场购进一批优质棉花,厂方技术员从A农场存储的优质棉花中随机抽取了100处棉花,分别测量了其纤维长度(单位:mm)的均值,收集到100个样本数据,并制成如下频数分布表:
长度(单位:mm) | [23,25) | [25,27) | [27,29) | [29,31) | [31,33) | [33,35) | [35,37) | [37,39] |
频数 | 4 | 9 | 16 | 24 | 18 | 14 | 10 | 5 |
(1)求这100个样本数据的平均数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c85481cd7e94130ef3aa05b4a39e79cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/671f43c79d612c93a6d160335e86e177.png)
(2)将收集到的数据绘成直方图可以认为这批棉花的纤维长度服从分布
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1290917c2c835b61384480b335cc1d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/594f4ea0d5f4a741355161af9eb34104.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d1316cf501a47b6d75594091d5f4a4.png)
①利用正态分布,求;
②纺织厂将A农场送来的这批优质棉进行二次检验,从中随机抽取20处测量其纤维均值yi(i=1,2…,20),数据如下:
y1 | y2 | y3 | y4 | y5 | y6 | y7 | y8 | y9 | y10 |
24.1 | 31.8 | 32.7 | 28.2 | 28.4 | 34.3 | 29.1 | 34.8 | 37.2 | 30.8 |
y11 | y12 | y13 | y14 | y15 | y16 | y17 | y18 | y19 | y20 |
30.6 | 25.2 | 32.9 | 27.1 | 35.9 | 28.9 | 33.9 | 29.5 | 35.0 | 29.9 |
若20个样本中纤维均值的频率不低于①中
即可判断该批优质棉花合格,否则认为农场运送时掺杂了次品,判断该批棉花不合格.按照此依据判断A农场送来的这批棉花是否为合格的优质棉花,并说明理由.
附:若,则
,
,
天数x | 1 | 2 | 3 | 4 | 5 | 6 |
抗体含量水平y | 5 | 10 | 26 | 50 | 96 | 195 |
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/5c434b92-e8ea-4036-af2c-f9df3df4c0d6.png?resizew=199)
(1)根据散点图判断,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b8b436f21307e26f87963d0e653fff0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4db1eeb94bd93c158cd90dadd4bcd37c.png)
(2)根据(1)的判断结果求出y关于x的回归方程,并预测该志愿者在注射疫苗后的第10天的抗体含量水平值;
(3)从这位志愿者的前6天的检测数据中随机抽取3天的数据作进一步的分析,求其中的y值小于50的天数X的分布列及数学期望.
参考数据:其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a7df4aa3f42a2c8d7f1f4c11c5dddde.png)
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
3.50 | 63.67 | 3.49 | 17.50 | 9.49 | 12.95 | 519.01 | 4023.87 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8254e5a17af8bf57847832fccf3e86c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e305ef27729a10ab9de0318e5374d406.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/11/31e7b8d5-2772-417b-80fe-a6bc29149695.png?resizew=159)
(1)从甲、乙两车间分别随机抽取2个零件,求甲车间至少一个零件合格且乙车间至少一个零件合格的概率;
(2)质检部门从甲车间8个零件中随机抽取4件进行检测,若至少2件合格,检测即可通过,若至少3件合格,检测即为良好,求甲车间在这次检测通过的条件下,获得检测良好的概率;
(3)若从甲、乙两车间12个零件中随机抽取2个零件,用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)定理证明:请用向量方法证明余弦定理(只需证明你写出的其中一个式子即可);
(3)定理应用:如图在四边形ABCD中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0bec02b6f97cbff99ba693116e2a980.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/814a5e193d8952447cabffb5f52c0348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71e90f9f4e44173888a54c624852064a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f08273d339dc5ddbb89aa67bb8205e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae8221601c7bd5c51fd520615581fa1.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
②求四边形ABCD的面积.
![](https://img.xkw.com/dksih/QBM/2022/5/9/2975642301808640/2978397887733760/STEM/63a371095226427da2dd46208040ca05.png?resizew=152)
6 . 某剧场的座位数量是固定的,管理人员统计了最近在该剧场举办的五场表演的票价(单位:元)和上座率
(上座人数与总座位数的比值)的数据,其中
,并根据统计数据得到如下的散点图:
(1)由散点图判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b447ac3d1a965572c31b6e4c18d4b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daae67e98fe0b96f9d949e8abb000c87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)根据(1)所求的回归方程,预测票价为多少时,剧场的门票收入最多.
参考数据:,
,
;设
,则
,
,
;
,
,
.
参考公式:对于一组数据,其回归直线
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)
月份![]() | 1 | 2 | 3 | 4 | 5 | 6 |
收入![]() | 6.8 | 8.6 | 16.1 | 19.6 | 28.1 | 40.0 |
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/8/f04e0350-445c-4b8b-90d3-9d4ba8c56353.png?resizew=172)
(1)根据散点图判断,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e136e7637543c8ae92c8dcd55b31924.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/581a6a69f1039aa12764eea5bf7ef405.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)根据(1)的结果及表中的数据,求出y关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)该车企为提高新能源汽车的安全性,近期配合中国汽车技术研究中心进行了包括跌落、追尾、多车碰撞等一系列安全试验项目,其中在实验场进行了一项甲、乙、丙三车同时去碰撞实验车的多车碰撞实验,测得实验车报废的概率为0.188,并且当只有一车碰撞实验车发生,实验车报废的概率为0.1,当有两车碰撞实验车发生,实验车报废的概率为0.2,由于各种因素,实验中甲乙丙三车碰撞实验车发生概率分别为0.7,0.5,0.4,且互不影响,求当三车同时碰撞实验车发生时实验车报废的概率.
参考数据:
![]() | ![]() | ![]() | ![]() | ![]() |
19.87 | 2.80 | 17.50 | 113.75 | 6.30 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c9607020fd3480f5cc027909a0a27fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5fd3eff2d3f8fd3c1e2a8c6b6a87e6.png)
参考公式:对于一组具有线性相关关系的数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc6c2985104b07933ffc17c2e5aab3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baab719e8fe66a8866e8971e426302dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238a8f3ee9f23642c22142b117404570.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b68acb9c8296e0dbf1ee49ab1768742.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5f28031b036e4a37be931d5ff28368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3432701dcc6000f25c5ea3a7937f2e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72656c882b2830eeb8053ec18c0088f.png)
在研究上面的【问题】时,小明和小宁分别得到了下面的【解法一】和【解法二】:
【解法一】由已知得方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13fa32c1e926f40a0722d106563777ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac9eb4f13a6ec140f7050e8d7dde52c.png)
由韦达定理得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c938b6e1b756e7320038f3437fc00522.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72656c882b2830eeb8053ec18c0088f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e785cde1d10e447e0bdd46f1ae065def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba4121651c3dcfe85bf13665250c2f43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a1a4025e867cbf2e7bc1749f0a0d78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72656c882b2830eeb8053ec18c0088f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee57417245352ff37564703524651589.png)
【解法二】由已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5f28031b036e4a37be931d5ff28368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/387ea55e8187d2dd5818e6a4e7ba5ab9.png)
令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd43ed419c7d59ac58907bce0a097afb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72656c882b2830eeb8053ec18c0088f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee57417245352ff37564703524651589.png)
参考以上解法,解答下面的问题:
(1)若关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf85d7e03bdb5fd044bbb52877411ef5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcbeb86ffdd71db9224d05e09a60f14e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53033cd3d75eb3bddb422b3108ba8573.png)
(2)若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aed1105dcd44df14cff24b5347effb17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70a3c53a296d8b450c835f8bd23d1238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0ea023914aff91c28a62cf70ba1b8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e4ed78f13799e014128b0472ea6f18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8ea893d278bb50bbe6a2b33b9c25be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75824b8a7a61eec9696329169e94a94a.png)
![](https://img.xkw.com/dksih/QBM/2022/3/30/2947492456726528/2948197200576512/STEM/b9cfbd73-d0b1-4a16-8714-94ccd0a6e26b.png?resizew=251)
(1)根据散点图判断,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a5b1c19e4c57f1d259f8269e551c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba6ef07026297f920ee8a61fba9436.png)
(2)①根据(1)的选择及表中数据,建立y关于x的回归方程(精确到0.001);
②若下一年销售额y需达到90亿元,预测下一年的研发资金投入量x约为多少亿元?
附:对于一组数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1e1872e1f4e195b307241343bae4b83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9ebbe62bcb439fe0b829c0f41318e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfdcb2ad70a5f88005a6b49c29106c0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa21830a185d89d13218784d3a9db6e.png)
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
20 | 30 | 3.2 | 900 | 300 | 3.78 | 1600 | 58.21 |
x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
y | 7 | 12 | 22 | 35 | 67 | 102 | 197 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9779ef5980d39d26bd92e4c504e86ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a3edbe89f552dd6cfd1abd462eef371.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cec12441802f71e803efaf2c62ee588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce64685821c3e55c07f151996ca8c3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/27/ce4493ec-0c2b-485d-bccd-c53533f97d02.png?resizew=211)
(2)根据(1)中求得的回归方程,求此回归模型投放量为5千辆时的残差
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c11887a27263903c7f3b141675ddd43.png)
参考数据:
![]() | ![]() | ![]() | ![]() |
63.14 | 1.56 | 2563 | 50.45 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39ff66375b8e2c21f2695655cd804782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3ce38a38bc80c9197c1b00b84c8743e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa41df1c1adbf9db409b34ec494c460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16137d6b120b9cc747f7cd4862fd631b.png)
参考公式:对于一组数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1c6aadc0129bf86f4fff9dcfb924b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/154100371e025fffe0ffae8be9567383.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a14e40329de36fc4a1a3f8fbfafda12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27e98f5555089692253c0c3f2ec0d92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74f38451023eaad524140b4ba939c375.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10a21112ac5c089fb55fb5f3e2a0e8f7.png)