生物统计习题及答案-2

生物统计与试验设计——作业二

彭继光 3090100060 植物保护

1. If a random variable X follows the normal distribution N(2, 25), let random variable Y =

X;2

5 ,z =*

2X −X

+,

1) what’s the expectation and variance of random variable Y ，what’s the distribution this random variable follows? E(Y)=E(

X;25

)=

E(X);25

=

2;25=0;

Var(Y)=Var(

X;25

)=Var(X

5)= 1

25σ2（X ）=25

25=1；

This random variable Y follows standard normal distribution Y~N(0,1)

2)deduce(推断) the expectation and variance of random vector z ，what ’s distribution it follows ？

E(z )=E *2X

−X +=[E(2X)E(−X)]=[2E(X)−E(X)]=*4−2

+

Var(z )=Var *2X

−X

+=Var(z) = E{[z －E(z )][z －E(z )]T }

=E ,[*2X −X +－*4−2+][*2X

−X +－*4−2

+]T-= E {*2X −4−X +2+*2X −4−X +2+T }

= E ,*2X −4−X +2

+[2X −4−X +2]-

= E [4(X −2)2−2(X −2)2−2(X −2)2(X −2)2]=[4E(X −2)2−2E(X −2)2

−2E(X −2)2E(X −2)2]

T =

X;μσ

~N(0,1) X =σ∙T +μ=5T+2 E(T)=0

E(X −2)2=E(5T +2−2)2=E(5T)2=25E(T 2)=25∫t 2:∞

;∞φ(t 2)dt =25∫t :∞

;∞√

2π

;t

2/2

dt =25

∴Var(z )= *4×25−2×25−2×2525+=*100−50

−5025+

∴z ~MVN (*4−2+,*100−50

−5025

+)

2. To study on the relationship between the lean meat (瘦肉量y) with eye muscle area (眼肌面积x 1)、leg meat (腿肉量x 2 )、waist meat (腰肉量x 3 ), a sample of 25 pigs was sampled, and trait investigation was conduced. The model y i =b 0+b 1x i1+b 2x i2+b 3x i3+e i was analyzed for the the multiple variable linear regression of lean meat on eye muscle meat, leg meat, waist meat. Suppose SSR =2

3.865，R 2=0.842，

b̂0(SE)= 0.857 (1.384),b̂1(SE)=0.0187(0.0296),b̂2(SE)=2.073(0.270)，b̂3(SE)=1.938

(0.513).

1) Please write out SAS statements for the step of procedure if the data set has been established;

date lean;

input eye leg waist @@;

card;

……

……

……

；

proc reg;

model lean= eye leg waist;

run;

2) Calculate the total sum of squares SSTO, residual sum squares SSE and adjusted determinant coefficient for the model R a2;

R2=SSR

SSTO SSTO=SSR

R2

=23.865

0.842

=28.343

SSTO=SSR+SSE SSE=SSTO-SSR=28.343-23.865=4.478

MSE=SSE/(n-p-1)= 4.478/(25-3-1)=0.213

MSTO=SSTO/(n-1)= 28.343/24=1.181

R a2=1−MSE/MSTO=1-0.213/1.181=0.820

3) Conduct statistical test on the linear relationship of lean meat on eye muscle meat, leg meat, waist meat, and give out appropriate statistical inference（统计推断）;

y i=0.857+0.0187x i1+2.073x i2+1.938x i3+e i

原假设H0：b1=b2=b3=0

备择假设H1：上式不成立

F∗=MSR

MSE =23.865

0.213

=112.042

F(0.05;p,n-p-1)=F(0.05;3,21)=3.07

∴F∗>F(0.05;3,21) 否定原假设，接受备择假设。

方差分析表明，F检验达到极显著水平。因而瘦肉量y与眼肌面积x1、腿肉量x2、腰肉量x3存在回归关系。大约82%的瘦肉量是由这三个产量构成因素决定的。

4) Conduct statistical test on the significance of each regression parameter, according to the results of statistical test, is it necessary to modify(修改) the model? why ?

上面第二题检验只是否定了所有自变量的回归参数均为零的假设H0：b1=b2=b3=0，但这并不能进一步判断认为每个自变量的回归参数均不为零。下面对每个自变量用t检验：