伤城文章网 >  > 2017-2018学年高中数学人教A版选修1-2:阶段质量检测(一) 统计案例 Word版含解析

2017-2018学年高中数学人教A版选修1-2:阶段质量检测(一) 统计案例 Word版含解析


阶段质量检测(一) 统计案例
(时间 120 分钟 满分 150 分) 一、选择题(本大题共 12 小题,每小题 5 分,共 60 分.在每小题给出的四个选项中, 只有一项是符合题目要求的) ^ ^ ^ ^ 1.对有线性相关关系的两个变量建立的回归直线方程 y = a + b x 中,回归系数b ( A.可以小于 0 C.能等于 0 B.大于 0 D.只能小于 0 )

^ ^ 解析:选 A ∵b =0 时,则 r=0,这时不具有线性相关关系,但b 可以大于 0 也可以 小于 0. ^ 2.每一吨铸铁成本 y(元)与铸件废品率 x%建立的回归方程 y =56+8x,下列说法正确 的是( )

A.废品率每增加 1%,成本每吨增加 64 元 B.废品率每增加 1%,成本每吨增加 8% C.废品率每增加 1%,成本每吨增加 8 元 D.如果废品率增加 1%,则每吨成本为 56 元 解析:选 C 根据回归方程知 y 是关于 x 的单调增函数,并且由系数知 x 每增加一个 单位,y 平均增加 8 个单位. 3.下表显示出样本中变量 y 随变量 x 变化的一组数据,由此判断它最可能是( x y A.线性函数模型 C.指数函数模型 4 14 5 18 6 19 7 20 8 23 9 25 10 28 )

B.二次函数模型 D.对数函数模型

解析:选 A 画出散点图(图略)可以得到这些样本点在某一条直线上或该直线附近,故 最可能是线性函数模型. 4.试验测得四组(x,y)的值为(1,2),(2,3),(3,4),(4,5),则 y 与 x 之间的回归直线方程 为( ) ^ ^ A. y =x+1 B. y =x+2 ^ ^ C. y =2x+1 D. y =x-1 ^ ^ 解析:选 A 由题意发现,(x,y)的四组值均满足 y =x+1,故 y =x+1 为回归直线方 程.

5.下列关于等高条形图说法正确的是( A.等高条形图表示高度相对的条形图 B.等高条形图表示的是分类变量的频数 C.等高条形图表示的是分类变量的百分比

)

D.等高条形图表示的是分类变量的实际高度 解析:选 C 由等高条形图的特点及性质进行判断. 6.根据一组样本数据(x1,y1),(x2,y2),…,(xn,yn)的散点图分析存在线性相关关系, ^ 求得其回归方程 y =0.85x-85.7,则在样本点(165,57)处的残差为( A.54.55 C.3.45 B.2.45 D.111.55 )

^ 解析:选 B 把 x=165 代入 y =0.85x-85.7,得 y=0.85×165-85.7=54.55, 由 57-54.55=2.45,故选 B. 7.有甲、乙两个班级进行数学考试,按照大于等于 85 分为优秀,85 分以下为非优秀 统计成绩,得到如下所示的列联表: 优秀 甲班 乙班 总计 10 c 非优秀 b 30 105 总计

2 已知在全部 105 人中随机抽取 1 人,成绩优秀的概率为 ,则下列说法正确的是( 7 A.列联表中 c 的值为 30,b 的值为 35 B.列联表中 c 的值为 15,b 的值为 50 C.根据列联表中的数据,若按 95%的可靠性要求,能认为“成绩与班级有关系”

)

D.根据列联表中的数据,若按 95%的可靠性要求,不能认为“成绩与班级有关系” 解析:选 C 由题意知,成绩优秀的学生数是 30,成绩非优秀的学生数是 75,所以 c =20,b=45,选项 A、B 错误.根据列联表中的数据,得到 K2= 105×?10×30-20×45?2 55×50×30×75

≈6.109>3.841,因此有 95%的把握认为“成绩与班级有关系”,选项 C 正确. 8.某考察团对全国 10 大城市进行职工人均工资水平 x(千元)与居民人均消费水平 y(千 ^ 元)统计调查,y 与 x 具有相关关系,回归方程为 y =0.66x+1.562,若某城市居民人均消 费水平为 7.675 千元,估计该城市人均消费额占人均工资收入的百分比约为( A.83% C.67% B.72% D.66% )

解析:选 A 将 y=7.675 代入回归方程,可计算得 x≈9.262,所以该城市人均消费 额占人均工资收入的百分比约为 7.675÷ 9.262≈0.83≈83%,即约为 83%. 9.为了研究男子的年龄与吸烟的关系,抽查了 100 个男子,按年龄超过和不超过 40 岁,吸烟量每天多于和不多于 20 支进行分组,如下表:

年龄 不超过 40 岁 吸烟量不多于 20 支/天 吸烟量多于 20 支/天 总计 10 60 25 40 50 超过 40 岁 15

总计

65

35 100

则在犯错误的概率不超过__________的前提下认为吸烟量与年龄有关( A.0.001 C.0.05 解析:选 A K2= B.0.01 D.没有理由 100×?50×25-10×15?2 ≈22.16>10.828, 65×35×60×40

)

所以我们在犯错误的概率不超过 0.001 的前提下认为吸烟量与年龄有关. 10.为了考察两个变量 x 和 y 之间的线性相关性,甲、乙两个同学各自独立做了 10 次 和 15 次试验,并且利用线性回归方法,求得回归直线为 l1 和 l2,已知在两人的试验中发现 对变量 x 的观测数据的平均值恰好相等, 都为 s, 对变量 y 的观测数据的平均值也恰好相等, 都为 t,那么下列说法正确的是( )

A.直线 l1 和直线 l2 有交点(s,t) B.直线 l1 和直线 l2 相交,但交点未必是点(s,t) C.直线 l1 和直线 l2 由于斜率相等,所以必定平行 D.直线 l1 和直线 l2 必定重合 解析:选 A l1 与 l2 都过样本中心( x , y ).

11.假设有两个分类变量 X 和 Y,它们的可能取值分别为{x1,x2}和{y1,y2},其 2×2 列联表如下: y1 x1 x2 总计 a c a+c y2 b d b+d 总计 a+b c+d a+b+c+d

对于以下数据,对同一样本能说明 X 与 Y 有关的可能性最大的一组为( A.a=9,b=8,c=7,d=6 B.a=9,b=7,c=6,d=8 C.a=8,b=6,c=9,d=7 D.a=6,b=7,c=8,d=9

)

解析:选 B 对于同一样本|ad-bc|越小,说明 X 与 Y 之间的关系越弱,|ad-bc|越大, 故检验知选 B. 12.两个分类变量 X 和 Y, 值域分别为{x1,x2}和{y1,y2}, 其样本频数分别是 a=10, b =21, c+d=35. 若 X 与 Y 有关系的可信程度不小于 97.5%, 则 c 等于( A.3 C.5 解析:选 A 列 2×2 列联表如下: x1 y1 y2 总计 故 K2 的观测值 k= 选 A. 二、填空题(本大题共 4 小题,每小题 5 分,共 20 分.请把正确答案填在题中的横线上) ^ 13.已知某车间加工零件的个数 x 与所花费时间 y(h)之间的线性回归方程为 y =0.01x +0.5,则加工 600 个零件大约需要________h. ^ 解析:当 x=600 时, y =0.01×600+0.5=6.5. 答案:6.5 14.若一组观测值(x1,y1),(x2,y2),…,(xn,yn)之间满足 yi=bxi+a+ei(i=1,2,…, n),若 ei 恒为 0,则 R2 为________. ^ 解析:ei 恒为 0,说明随机误差总为 0,于是 yi= y ,故 R2=1. 答案:1 15.下列是关于出生男婴与女婴调查的列联表 晚上 男婴 女婴 总计 45 E 98 白天 A 35 D 总计 B C 180 10 c 10+c x2 21 d 21+d 总计 31 35 66 B.4 D.6 )

66×[10?35-c?-21c]2 ≥5.024. 把选项 A, B, C, D 代入验证可知 31×35×?10+c??56-c?

那么 A=______,B=______,C______,D=________,E=________. 解析:∵45+E=98,∴E=53, ∵E+35=C,∴C=88,∵98+D=180,∴D=82, ∵A+35=D,∴A=47,∵45+A=B,∴B=92. 答案:47 92 88 82 53

16.已知 x,y 之间的一组数据如表,对于表中数据,甲、乙两同学给出的拟合直线分 1 1 1 别为 l1:y= x+1 与 l2:y= x+ ,利用最小二乘法判断拟合程度更好的直线是________. 3 2 2 x y 1 1 3 2 6 3 7 4 8 5

1 解析:用 y= x+1 作为拟合直线时,所得 y 的实际值与 y 的估计值的差的平方和为: 3 4 10 11 7 1 1 1- ?2+(2-2)2+(3-3)2+?4- ?2+?5- ?2= .用 y= x+ 作为拟合直线时,所 S1=? 3? ? 3? 3 ? 3? ? 2 2 7? 2 2 得 y 的实际值与 y 的估计值的差的平方和为:S2=(1-1)2+(2-2)2+? ?3-2? +(4-4) +

?5-9?2=1. ? 2? 2
1 1 因为 S2<S1,故用直线 l2:y= x+ ,拟合程度更好. 2 2 1 1 答案:y= x+ 2 2 三、解答题(本大题共 6 小题,共 70 分,解答时写出必要的文字说明、证明过程或演算 步骤) 17.(本小题满分 10 分)对某校小学生进行心理障碍测试得如下列联表:(其中焦虑、说 谎、懒惰都是心理障碍) 焦虑 女生 男生 总计 5 20 25 说谎 10 10 20 懒惰 15 50 65 总计 30 80 110

试说明在这三种心理障碍中哪一种与性别关系最大?
2 2 解:对于上述三种心理障碍分别构造三个随机变量 K2 1,K2,K3,由表中数据可得

K2 1= K2 2=

110×?5×60-25×20?2 ≈0.863, 30×80×25×85 110×?10×70-20×10?2 ≈6.366, 30×80×20×90

K2 3=

110×?15×30-15×50?2 ≈1.410. 30×80×65×45

因为 K2 2的值最大,所以说谎与性别关系最大. 18.(本小题满分 12 分)有人统计一个省的 6 个城市某一年的人均国内生产总值(人均 GDP)x 和这一年各城市患白血病的儿童数量 y,其数据如下表所示: 人均 GDP x/万元 患白血病的儿童数量 y/人 10 351 8 312 6 207 4 175 3 132 1 180

(1)画出散点图,并判断是否线性相关; (2)求 y 与 x 之间的回归方程. 解:(1)作散点图(如下图所示).

由散点图可知 y 与 x 具有线性相关关系. ^ ^ (2)将数据代入公式,可得b ≈23.253, a ≈102.151. ^ 故 y 与 x 之间的线性回归方程是 y =23.253x+102.151. 19.(本小题满分 12 分)某校在两个班进行教学方式对比试验,两个月后进行了一次检 测,试验班与对照班成绩统计如下表所示(单位:人): 80 及 80 分以上 试验班 对照班 总计 35 20 55 80 分以下 15 m 45 总计 50 50 n

(1)求 m,n; (2)能否在犯错误的概率不超过 0.005 的情况下认为教学方式与成绩有关系? 解:(1)m=45-15=30,n=50+50=100. (2)由表中的数据,得 K2 的观测值为 k= 100×?35×30-15×20?2 ≈9.091. 50×50×55×45

因为 9.091>7.879,所以能在犯错误的概率不超过 0.005 的前提下认为教学方式与 成绩有关系.

20.(本小题满分 12 分)某工厂用甲、乙两种不同工艺生产一大批同一种零件,零件尺 寸均在[21.7,22.3](单位:cm)之间,把零件尺寸在 [21.9,22.1)的记为一等品,尺寸在 [21.8,21.9)∪[22.1,22.2)的记为二等品,尺寸在[21.7,21.8)∪[22.2,22.3]的记为三 等品,现从甲、乙工艺生产的零件中各随机抽取 100 件产品,所得零件尺寸的频率分布直 方图如图所示:

(1)根据上述数据完成下列 2×2 列联表, 根据此数据你认为选择不同的工艺与生产出一 等品是否有关? 甲工艺 一等品 非一等品 总计 乙工艺 总计

附: P(K2≥k0) k0 n?ad-bc?2 K= ?a+b??c+d??a+c??b+d?
2

0.10 2.706

0.05 3.841

0.01 6.635

(2)以上述各种产品的频率作为各种产品发生的概率,若一等品、二等品、三等品的单 件利润分别为 30 元、20 元、15 元,你认为以后该工厂应该选择哪种工艺生产该种零件? 请说明理由. 解:(1)2×2 列联表如下 甲工艺 一等品 50 乙工艺 60 总计 110

非一等品 总计

50 100

40 100

90 200

K2=

200×?50×40-60×50?2 ≈2.02<2.706,所以没有理由认为选择不同的工艺与生 110×90×100×100

产出一等品有关. (2)由题知运用甲工艺生产单件产品的利润 X 的分布列为 X P 30 0.5 20 0. 3 15 0.2

X 的数学期望为 E(X)=30×0.5+20×0.3+15×0.2=24,X 的方差为 D(X)=(30 -24)2×0.5+(20-24)2×0.3+(15-24)2×0.2=39. 乙工艺生产单件产品的利润 Y 的分布列为 Y P 30 0.6 20 0. 1 15 0.3

Y 的数学期望为 E(Y)=30×0.6+20×0.1+15×0.3=24.5, Y 的方差为 D(Y)=(30-24. 5)2×0. 6+(20-24. 5)2×0. 1+(15-24. 5)2×0. 3=47. 25. 由上述结果可以看出 D(X)<D(Y),即甲工艺波动小,虽然 E(X)<E(Y),但相差不大,所 以以后选择甲工艺. 21.(本小题满分 12 分)为调查某地区老年人是否需要志愿者提供帮助,用简单随机抽 样的方法从该地区调查了 500 位老年人,结果如下: 性别 是否需要志愿者 需要 不需要 男 40 160 女 30 270

P(K2≥k0) k0

0.05 3.841

0.01 6.635

0.001 10.828

附:K2 的观测值 k=

n?ad-bc?2 . ?a+b??c+d??a+c??b+d?

(1)估计该地区老年人中,需要志愿者提供帮助的老年人的比例; (2)在犯错误的概率不超过 0.01 的前提下是否可认为该地区的老年人是否需要志愿者 提供帮助与性别有关?

(3)根据(2)的结论,能否提出更好的调查方法来估计该地区的老年人中,需要志愿者提 供帮助的老年人的比例?请说明理由. 解:(1)调查的 500 位老人中有 70 位需要志愿者提供帮助,因此该地区老年人中,需要 帮助的老年人的比例的估算值为 (2)随机变量 K2 的观测值 500×?40×270-30×160?2 k= ≈9.967. 200×300×70×430 由于 9.967>6.635,因此,在犯错误的概率不超过 0.01 的前提下认为该地区的老年 人是否需要志愿者提供帮助与性别有关. (3)由(2)的结论知,该地区的老年人是否需要帮助与性别有关,并且从样本数据中能看 出该地区男性老年人与女性老年人中需要帮助的比例有明显差异,因此在调查时,先确定 该地区老年人中男、女的比例,再把老年人分成男、女两层,并且采用分层抽样方法比采 用简单随机抽样的方法更好. 22.(本小题满分 12 分)某市为了对学生的数理(数学与物理)学习能力进行分析,从 10 000 名学生中随机抽出 100 位学生的数理综合学习能力等级分数(6 分制)作为样本,分数频 数分布如下表: 等级得分 人数 (0,1] 3 (1,2] 17 (2, 3] 30 (3,4] 30 (4,5] 17 (5,6] 3 70 =14%. 500

(1)如果以能力等级分数大于 4 分作为良好的标准,从样本中任意抽取 2 名学生,求恰 有 1 名学生为良好的概率. (2)统计方法中,同一组数据常用该组区间的中点值(例如区间(1,2]的中点值为 1.5)作 为代表: ①据此,计算这 100 名学生数理学习能力等级分数的期望 μ 及标准差 σ(精确到 0.1); ②若总体服从正态分布,以样本估计总体,估计该市这 10 000 名学生中数理学习能力 等级在(1.9,4.1)范围内的人数. (3)从这 10 000 名学生中任意抽取 5 名同学,他们数学与物理单科学习能力等级分数如 下表: x(数学学习能力) y(物理学习能力) 2 1.5 3 3 4 4.5 5 5 6 6

①请画出上表数据的散点图; ^ ^ ^ ②请根据上表提供的数据,用最小二乘法求出 y 关于 x 的线性回归方程 y = b x+ a (附参 考数据: 129≈11.4). 解:(1)样本中学生为良好的人数为 20 人.故从样本中任意抽取 2 名学生,则仅有 1 名
1 C1 32 20×C80 学生为良好的概率为 = . 2 C100 99

(2)①总体数据的期望约为:μ=0.5×0.03+1.5×0.17+2.5×0.30+3.5×0.30 +4.5×0.17+5.5×0.03=3.0, 标准差 σ=[(0.5-3)2× 0.03+(1.5-3)2× 0.17+(2.5-3)2× 0.3+(3.5-3)2× 0.3 1 +(4.5-3)2× 0.17+(5.5-3)2× 0.03] = 1.29≈1.1, 2 ②由于 μ=3,σ=1.1 当 x∈(1.9,4.1)时,即 x∈(μ-σ,μ+σ), 故数理学习能力等级分数在(1.9,4.1)范围中的概率为 0.682 6. 数理习能力等级分数在(1.9,4.1)范围中的学生的人数约为 10 000×0.682 6=6 826 人. (3)①数据的散点图如图:

^ ^ ^ ②设线性回归方程为 y =b x+ a ,则

^ b=

i=1

?xiyi-5 x ? x2 i -5 x
5 2

5

y ^ ^ =1.1, a = y - b x =-0.4.

i=1

^ 故回归直线方程为 y =1.1x-0.4.


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