#5.16
q<-4;n<-355;alpha<-0.05
p<-c(.3,.33,.16,.07,.14)
diag(sigma)<-p*(1-p)
UCI<-p+sqrt(qchisq(1-alpha, q)*diag(sigma)/n)
LCI<-p-sqrt(qchisq(1-alpha, q)*diag(sigma)/n
The simultaneous 95% confidence intervals are
p1: [0.22508365, 0.3749163]
p2: [0.25312921, 0.4068708]
p3: [0.10006692, 0.2199331]
p4: [0.02828834, 0.1117117]
p5: [0.08327426, 0.1967257]
The simultaneous 95% confidence interval that allows a comparison of the Bank of Shorewood
with Bank B is (-0.1596662, 0.0996662). Since the interval contains 0, there is no significant
difference between them.
#5.20
280
278
276
274
x2
282
284
a)
190
192
194
x1
196
198
From the plot, I found that c (190,275) lies inside the 95% confidence ellipse. So there is no
evidence to reject the hypothesis that they are plausible values for the mean tail length and
mean wing length for the female birds.
b) Simultaneous 95% T2 intervals are:
x1: (189.4217, 197.8227)
x2: (274.2564, 285.2992)
95% Bonferroni intervals:
x1: (189.8216, 197.4229)
x2: (274.7819, 284.7736)
T2 intervals are wider than the Bonferroni intervals, since the Bonferroni test is a conservative
test.
c) Q-Q plots:
The result Shapiro-Wilk normality tests:
X1
X2
Test statistics
0.9698
0.9813
p-value
0.2857
0.6726
From both Q-Q plots and Shapiro-Wilk tests, there are no evidences to reject the normality
assumption for x1 and x2.
The scatter diagram is
320
300
320
300
260
240
280
Wing length
260
240
280
170
180
190
170
180
190
200
210
220
210
220
Tail length
200
The chi-square plot of the ordered distances is:
From the scatter diagram, I found that points can form an ellipse. The points on the chisquare plot can approximately form a line. So I think the bivariate normal distribution is a viable
population model.
#5.22 (I used the 25 observations; answers may vary if you use all the observations)
It is obvious that observations 9 and 21 are outliers. After removing the outliers, we can see
from Q-Q plots that each variable appears to be normally distributed. But each pair of variables
are not apparently bivariate normal.
b) The Bonferroni intervals:
x1: (9.789733, 15.33027)
x2: (5.777122, 10.54528)
x3: (8.646243, 12.44256)
The T2 intervals:
x1: (9.159708, 15.96029)
x2: (5.234926, 11.08747)
x3: (8.214557, 12.87424)
The T2 intervals are wider than the Bonferroni intervals.
#6.17
word <- read.table(file = "http://www.public.iastate.edu/~maitra/stat501/datasets/word.dat",
header = F)
n<-nrow(word);q<-ncol(word)
mx<-mean(word)
S<-cov(word)
C<-matrix(c(-1,-1,1,1,1,-1,1,-1,1,-1,-1,1),byrow=T,nrow=3)
a)
T2<-n*(t(C%*%mx))%*% solve(C%*%S%*%t(C)) %*% (C%*%mx)
# The value for T2 is 135.8543.
test<-(n-1)*(q-1)/(n-q+1)*qf(p=0.95,df1=q-1,df2=n-q+1)
#T2>test. So we have significant evident to reject the null hypotheses that they have no
treatment effects.
b)
LCI<-C%*%mx-sqrt(test/n* diag(C%*%S%*%t(C)) )
UCI<-C%*%mx+sqrt(test/n* diag(C%*%S%*%t(C)) )
CI<-cbind(LCI,UCI)
dimnames(CI) <- list(c("format","type","interation"),c("LCI", "UCI"))
LCI
UCI
format -411.56737 -186.83888
type
124.97937 280.36438
interation -75.05883 32.40258
Since 0 is included in the interval for the interaction, there is no significant evidence to proof the
interaction effect. But there are significant evidence to proof the format effect and type effect.
c)
From b) there is no interaction effect so M model of numerical cognition is supported in this
experiment.
d)
The chi-square Q-Q plot appears approximately a straight line except for the largest
three points. The test statistic for the correlation test is r= 0.980392 and p-value=0.405. So
there is no sufficient evidence to reject the multivariate normal assumption.
#6.22
a)
The value for T2 is 96.37322 and the p-value is less than 0.0001. We can reject the null
hypothesis so there are significant gender differences. The linear combination most responsible
for the rejecting is (-95.600, 6.145, 5.737, -0.762).
b)
The 95% simultaneous confidence intervals:
X1: ( -0.002523361, 0.1697234)
X2: (-1.163483457, 1.4650835)
X3: (0.868742824, 1.8760572)
X4: (5.387340281, 17.1438597)
The Bonferroni intervals:
X1: (0.01621484, 0.1509852)
X2: (-0.87752962, 1.1791296)
X3: (0.97832550, 1.7664745)
X4: (6.66629645, 15.8649035)
Again, the Bonferroni intervals are narrower.
c)
The data were collected from graduate-student volunteers and thus they do not represent a
random sample. So the sample is not representative for the whole population. So the inference
from this study cannot be generalized to a larger population.