2023年应用统计方法大作业

例3-7

在某项实验中，测得变量y与因素x数据如表1所示。试建立适当的y与因

素x的回归方程（。=0。1）o

表3-14例3-7实测数据

X23457810111415161879

yI0/010/1109.93110.49110.5911/1111111

6.428.9.5809.510.610.0.761.2

209

解:绘制散点图，如图1所示:

y

图1

从图1中可看出，以下三种曲线方程的曲线图都与散点图接近，因此都可以

作为曲线回归的选择对象。

（I）y=Bo+Bi6.

（2）y=@）+£Jgx

⑶y=氏+x

现对方案1和方案2方案3进行求解分析，通过对S2残作比较，S2或小这则

回归方程交优

1.方案1

选取曲线回归（1）进行求解。令x=4,应用EXCEL进行相应解决算得数

据，列入表1。

•~i

Xx=Vxyx—XYi-y(x'-x')(yi一/

21.106.42-1.-3.5.

31.108.2-1.-1.2.

42109.58-1.-o.0.

52.109.5-0.-0.0.

72.110-0.0.-0.

82.109.93-0.-0.0.

103.16227766110.490.0.0.

113.31662479110.590.0.0.

143.110.60.699227110.0.

153.110.90.0.0.

164110.760.0.0.

184.1111.202310411.1.

194.111.21.1.1.

表1方案1数据解决计算

39.55159361429.1713.93894078

109.93

平均3.

62

15

平方和13211.6670341421.2105076944.12800113

7138.7

由表2得:

13_2

=11.66703

Z=1

13__

U=Z(^-?)(X-j)=13.93894

<=i

=21.210508

由此得:

13.938941

=1.1947

»-11.667034

A-A___

用=y一4/=109.9362-1.1947X3.042430=106.3014

故所求的回归方程为:

=106.3014+1.1947%

进行变量还原得回归方程：

^=106.3014+1.1947^

检查假设H°i：4=0.

。而尸/?,/；v=1.1947x13.9389=16.6528

S媒=&一够=21.2105—16.6528=4.5577

16.6528

=40.1915

4.5577/11

对给定的a=0.01,查表（附表5）得临界值丸=9.65。由于F>4,检

查效果显著，所以拒绝Hoi,即回归方程故意义。

2.方案2

表2方案2数据解决计算

1~r

XX'=1gxYX-Xyi-y(x'-x')(yi-y)

106.4202—0.61-3.5162

20.3010302.168071

306595oo

30.477121108.202300-0.440504-1.7362000.764803

10-0.3—0.3562

40.6020600.112404

9.5800001556500

-0.21865

50.698970109.500000-0.4362000.095377

5

70.845098110.000000-0.0725270.063800-0.004627

109.93000-0.0145

80.903090-0.0062000.000090

035

0.55380

101.000000110.4900000.0823750.045619

0

111.041393110.5900000.1237680.6538000.080919

141.146128110.6000000.2285030.6638000.151680

1.17609

15110.9000000.2584660.9638000.249110

1

161.204120110.7600000.2864950.8238000.236015

1.06380

181.255273111.0000000.3376480.359189

0

1.27875111.20

190.3611291.2638000.456394

42300

1429.1700

总和11.929127———

00

平均0.917625109.936154———

平方和——1.19471721.21.5104.715045

=1.1947

2

y.v=Z(x-y)।=21.210508

i=\

13_、/

K=X卜;—％')(y—y)=4.7150

/=1

由此得:

4.7150

=3.9466

1.1947

A_A___

&=y-4％'=109.9362-3.9466x0.9176=106.3147

故所求回归方程:

y=106.3147+3.9466》

进行变量还原得回归方程：

y=106.3147+3.94661gx

检查假设：Ho」优=0

A

S合四L=3.9466x4.7150=18.6083

T=21.2105-18.6083=2.6022

18.6083

=78.6605

2.6022/11

对给定的a=0.01,查F(1,11)表得到临界值入=9.65.由于F＞人,检查效

果显著，所以拒绝即回归方程故意义。

3.方案3

选取曲线回归(3)求解。令x=L应用EXCEL可算的数据,列入表30

X

表3方案3数据解决计算

XX=Vxyx-X力一旷(x-x)(yi-50

20.5106.420.-3.-1.

30.108.20.-1.-0.

40.25109.580.-0.-0.

50.2109.50.-0.-0.

70.110-0.0.-0.0009515

80.125109.93-0.-0.0.

100.1110.49-0.0.-0.

110.110.59-0.0.-0.

140.110.6-0.0.一0.

150.110.9-0.0.-0.

160.0625110.76-0.0.-0.

180.111-0.1.-0.

190.111.2-0.105128571.-0.

21.210507

Z2.050881941429.170.-2.

69

109.93

平均0.

615

15713

平方和0.

8.66

由表3得

13_

J=Z（x；-％）2=0.2137

i=l

13_

4=Z（y7）2=212105

Z=1

13__

A.=Z（%；一入）（必一月=一2.1011

**y

/=i

由此得：

I-?1011

q==_9832（）

1/.0.2137

XX

Bo=y~P^x=\09.9362+9.8320x0.1578=111.4877

故所求的回归方程为：

y=111.4877-9.8320%

进行变量还原得回归方程:

!11.4877-9.8320-

x

检查假设Hoi：4=0.

陷=耳(=(-9.8320)x(-2.1011)=20.6580

枭=Zvv-5^=21.2105-20.6580=0.5525

20.6580

F=411.2905

SQll0.5525/11

对给定的a=0.01,查F(l,11)表(附表5)得临界值2=9.65。由于F>2,

检查效果显著，所以拒绝Ho-即回归方程故意义。

表4

方案回归方程s晟F

1y=106.3014+1.194764.557740.1915

2y=106.3147+3.9466x2.602278.6605

3j=ll1.4877-9.8320-0.5525411.2905

由表4,方案3的残差平方和是最小的，因而其回归方程是最优的，拟合效果

是最佳的，方案2次之，方案1最差。

第三章课后题

5、研究平炉炼钢的效率y与出钢量(X1)和FeO(%)的关系，测得数据如下:

101

X1115.396.556.9101.0102.987.9109.8

1.403.4

x214.214.614.914.918.213.213.520.013.0

y83.578.073.091.483.482.084.080.088.0

8

xl110.680.393.088.0108.989.5104.4101.9

8.0

x215.312.914.718.118.115.418.313.812.2

y86.581.088.685.785.781.979.189.980.6

(1)建立y关于X],%的线性回归方程

(2)检查所建方程是否故意义(a=0.10)

(3)检查X1,乙是否对y有显著影响(a=0.10)

(4)假如有对y影响不显著的变量，将其去掉再建立一元回归方程

表1数据预解决计算

x

i%2yxiX2x,x2X]%2丁V

11

114.283.513294.09201.641637.269627.551185.76972.25

5.3

296.514.6789312.25213.161408.975271138.86084

14.

356.9733237.61222.01847.814153.71087.75329

9

9

410114.910201222.011504.99231.41361.868353.96

1.4

1011058

583.4331.241872.788581.861517.886955.56

2.98.28.41

687.913.2827726.41174.241160.287207.81082.46724

7101.413.58410281.96182.251368.98517.611347056

101205

820804002196878416006400

9.86.04

9103.4138810691.51691344.29099.211447744

6

12232.3

10110.615.386.5234.091692.189566.91323.457482.25

6

1180.312.9816448.09166.411035.876504.31044.96561

129314.788.68649216.091367.18239.81302.427849.96

138816.481.57744268.961443.271721336.66642.25

7344.4

148818.185.77744327.611592.87541.61551.17

9

10

1515.481.911859.21237.161677.068918.911261.266707.61

8.9

6256.8

1689.518.379.18010.25334.891637.857079.451447.53

1

17104.413.889.910899.36190.441440.729385.561240.628082.01

101038

1812.280.6148.841243.188213.14983.326496.36

1.93.61

1739.1171351453522712504

和273.64240.0426470.99

7498.19.211.7743.611.51

83.2

平均96.6515.2

3

⑴

设g=bo+bixi+b2X2

由表1得

18

InNx2ii-18x2i=171359.21--x1739.72=3217.205

i=l18

181

112=Vxiix.2-18x1x2=26470.99--x1739.7x273.6=27.55=I21

tT18

122=£18

X2i2-18x22=4240.04--X273.62=81.32

i=l18

18

liy=^yiXii-18xiy=145351.77-—X1739.7X1498.1=560.405

i=l