1 . 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
,
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb398137779190b35492d9f06d5fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c214cc074cf24aa90f2dfb01de102a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/285ebeb817033552af0f67d73496de4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/251f75fa86b905a2616ec2762d7f7aa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5849eaa8f3a7122ee92fb55f4b362fc.png)
(Ⅰ)从该地区的蜻蜓中随机捕捉一只,求这只蜻蜓的翼长在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fe763a71976c3a8de837e1b4a3f335.png)
(Ⅱ)记该地区蜻蜓的翼长为随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d768b45940249b92c7de69ca3bfb410.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e34bde9ce11f753f3e3631fbd0112fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08e2ceedae2559a314a877439b8de231.png)
(Ⅲ)在(Ⅱ)的条件下,从该地区的蜻蜓中随机捕捉3只,记这3只中翼长在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a36b5f3a3b485fa37df9c6575d499736.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
注:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db5f0cb8603471163cf0e3938a5039a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/008c275c8688fa0821245614f54a12cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/256181dd02e41f3a9eadf1de097f472e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd5de2f466b858a78245dffd6809928.png)
(1)实际情景
大家在农贸市场购物时,通常会发现在农贸市场管理处的窗口,都会有一个电子秤,以方便老百姓核对斤两与价格,这是为了放置商贩在售卖中缺斤少两.国家有关部门也对缺斤少两作出了相应的定义:比如粮食、蔬菜、水果或每公斤不高于6元的食品,称重范围等于或小于1公斤的其负偏差不能超过20克;大于1公斤,等于或小于2公斤的不能超过40克;大于2公斤,等于或小于4公斤的不能超过80克;大于4公斤,等于或小于25公斤的不能超过100克.其实在网购时也有缺斤少两的问题时,由此产生经济纠纷.
(2)提出问题
李师傅想利用手机在某外卖平台商家
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)分析问题
是否缺斤少两,要从统计学的角度取分析,一般来说,每份水果的质量应该服从正态分布,根据所学的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45d81fb351bc6c693fb0a114152b2d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
2.收集数据
李师傅决定师傅从2022年3月1日至6月8日连续100天,每天都在平台上购买一份水果,并将水果的重量记录如下:
983 | 993 | 1013 | 963 | 973 | 963 | 993 | 993 | 993 | 953 |
1023 | 1023 | 993 | 1013 | 873 | 1014 | 1003 | 1033 | 1003 | 993 |
1032 | 1023 | 993 | 993 | 1011 | 993 | 1003 | 943 | 973 | 1013 |
1013 | 973 | 1023 | 1013 | 1023 | 1003 | 963 | 943 | 963 | 1023 |
1023 | 913 | 1023 | 983 | 1013 | 973 | 973 | 993 | 1013 | 1013 |
1023 | 1023 | 953 | 873 | 973 | 993 | 823 | 993 | 1023 | 1013 |
1003 | 1022 | 1023 | 973 | 953 | 993 | 1023 | 993 | 943 | 1023 |
993 | 1001 | 983 | 953 | 993 | 1023 | 993 | 1023 | 983 | 1003 |
893 | 1013 | 973 | 933 | 1013 | 1043 | 863 | 1013 | 973 | 963 |
843 | 1013 | 983 | 1023 | 943 | 883 | 773 | 1023 | 983 | 973 |
983 | 993 | 1013 | 963 | 973 | 963 | 993 | 993 | 993 | 953 |
1023 | 1023 | 993 | 1013 | 873 | 1014 | 1003 | 1033 | 1003 | 993 |
上述数据较多,我们利用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c4f57c68f4ff96c66c5402e6932c6c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb9687c8a02a6ea2bf65232fcd288ac5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/046680467b09c8bf4d9e6bbf09f5eb1e.png)
4.问题解决
水平的重量理论上重量的分布应该近似服从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b50b3d7e66c214980ea5d921aa5b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa106b9769ea53c8e8242e53cff92e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09e4454be408f003241ecd98c9022d4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2c4d12b3a705daab723ab243b6cc88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45d81fb351bc6c693fb0a114152b2d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09e4454be408f003241ecd98c9022d4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ad022a0bace017e8f560a0e558970e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
5.检验模型
本案例中由于李师傅所取样本的容量为100,样本数量较少,为了减少误差,可以增加样本的容量,以保证所得结论的准确性.
6.拓展延伸
正态分布的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45d81fb351bc6c693fb0a114152b2d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)通过长期调查分析可知,该养殖场家禽血液中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f869a23427e1e58d980be6294bd5f250.png)
(i)若其中一个养殖棚有1000只家禽,估计其中血液
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de2f6537a05597d665e3b38545d29cc2.png)
(ii)在统计学中,把发生概率小于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edb1e0c3e2fb29b0b35d51d22a5710d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec280dc00533ecaab6713140566d59de.png)
参考数据:
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25591e317926f11b3dc87550a2ad4604.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53c1ed67167078ea4f5f1ee53ee14164.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6fba95264b21f99bae6e520ff071e93.png)