生統 96-1 HW#11&12
慈濟醫資系 2008/01/09 P.1
HW#11 Solution
HW11 source: Principles of Biostatistics, 2nd ed., M. Pagano and K. Gauvreau, Duxbury, 2000
1. Exercise 12.8 (a), (b), (c), (d)。（p. 299）
2. Exercise 12.9 (a), (b), (c), (d), (e)。（p. 299）
12.8
a. To test H0 :μ1=μ2=μ3=μ4 , we must first calculate estimates of the
within-groups and between-groups variances.
sW2 = [(n1 – 1) s12 + (n2 – 1) s22 + (n3 – 1) s32 + (n4 – 1) s42]/(n – 4) = 1.75
Since x = (n1 x 1 + n2 x 2 + n3 x 3 + n4 x 4)/n = 5.58,
the between-groups variability:
sB2 = [n1( x 1 – x )2 + n2( x 2 – x )2 + n3( x 3 – x )2 + n4( x 4 – x )2]/(4 – 1) = 19.06.
The test statistic is F = sB2/ sW2 = 19.06/1.75 = 10.89 ~ Fdf1 , df2 = F3, 1494, under H0.
The p-value < 0.001, therefore, we reject the null hypothesis.
b. We conclude that mean LDL cholesterol level is not the same for all four groups.
c. In order to use the one-way ANOVA, the four populations must be at least
approximately normally distributed, and their variances must all be the same.
d. It is necessary to take an additional step in this analysis. We have concluded that
the mean LDL levels are not the same for all four groups, but we cannot yet say
which group means differ from which others. In order to do this, we need to use the
Bonferroni method of multiple comparisons. (It would tell us that patients with
intermittent claudication and those with minor asymptomatic disease have higher
mean LDL cholesterol levels than patients with no disease.)
12.9
a. The average minutes of individual therapy per session is longest for the private
not-for-profit centers and shortest for the for-profit centers; the average minutes of
group therapy is longest for the for-profit centers and shortest for the public centers.
b. Each individual therapy, the three confidence intervals all overlap. Therefore, there
is nothing to suggest that the populations means are not identical. (This is not a
formal test, however.) The same is true for group therapy.
c. To test H0 :μ1=μ2=μ3 individual therapy for each type of center, we first
calculate estimates of the within-groups and between-groups variances.
sW2 = [(n1 – 1) s12 + (n2 – 1) s22 + (n3 – 1) s32]/(n – 3) = 135.4
Since x = (n1 x 1 + n2 x 2 + n3 x 3)/n = 53.88,
the between-groups variability:
sB2 = [n1( x 1 – x )2 + n2( x 2 – x )2 + n3( x 3 – x )2]/(3 – 1) = 515.8.
The test statistic is F = sB2/ sW2 = 515.8/135.4 = 3.81 ~ Fdf1 , df2 = F2, 515, under H0.
The 0.01 < p-value < 0.025, therefore, we reject H0 at 0.05 level and conclude that
mean minutes of individual therapy per session are not the same for all three types
of centers.
生統 96-1 HW#11&12
慈濟醫資系 2008/01/09 P.2
To apply the Bonferroni method of multiple comparisons, we conduct three pairwise
tests at the 0.05/3 = 0.0167 level of significance. The test statistics are:
t12 = [( x1  x2 )  (1  2 )]/ sW2 (1/ n1 )  (1/ n2 )
= [(49.46  54.76)  0]/ 135.4[(1/ 37)  (1/ 312)] = -2.62
t13 = [(49.46  53.25)  0]/ 135.4[(1/ 37)  (1/169)] = -1.79
t23 = [(54.76  53.25)  0]/ 135.4[(1/ 312)  (1/169)] = 1.36
All test statistics have a t distribution with 515 df. The comparison of private
for-profit and not-for-profit centers results in p = 0.008; the p-values for the other
two comparisons are both greater than 0.0167. Therefore, we conclude that the
mean minutes of individual therapy per session is shorter for for-profit centers than
for not-for-profit centers.
d. To test H0 :μ1=μ2=μ3 group therapy for each type of center, we again calculate
estimates of the within-groups and between-groups variances.
sW2 = [(n1 – 1) s12 + (n2 – 1) s22 + (n3 – 1) s32]/(n – 3) = 947.7
Since x = (n1 x 1 + n2 x 2 + n3 x 3)/n = 97.60,
the between-groups variability:
sB2 = [n1( x 1 – x )2 + n2( x 2 – x )2 + n3( x 3 – x )2]/(3 – 1) = 2159.2.
The test statistic is F = sB2/ sW2 = 2159.2/947.7 = 2.28 ~ F2, 488, under H0.
The p-value > 0.10, therefore, we cannot reject H0 at 0.05 level and conclude that
mean minutes of group therapy per session are the same.
e. While private for-profit centers have shorter individual therapy sessions than
not-for-profit centers, on average, there are no significant differences in the length
of group therapy sessions.
生統 96-1 HW#11&12
慈濟醫資系 2008/01/09 P.3
HW#12
HW12 source: Principles of Biostatistics, 2nd ed., M. Pagano and K. Gauvreau, Duxbury, 2000
1. Exercise 17.5 (a), (b), (c), (d), (e), (f), (g)。（p. 412）
2. Exercise 18.9 (b), (c), (d), (e)。（p. 444）
3. Exercise 18.11 (a), (b), (c), (d)。
（p. 445, B-10）
17.5
(a)
11.00
Ch o le s te ro l Le ve l
10.00
9.00
8.00
7.00
6.00
R S q Line a r = 0.422
5.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
Trig lyc e rid e Le ve l
(b) There appears to be a slight tendency for cholesterol to increase as triglyceride
increases.
1 n
(c) Pearson’s coefficient of correlation r =
 ( xi  x )( yi  y ) sx s y
n  1 i 1
= 34.90/(10 – 1)(1.563)(3.818) = 0.650.
(d) Test statistic T = r (n  2) /(1  r 2 ) = .650 (10  2) /(1  .652 ) = 2.42
Under H0, the quantity T ~ t8, if X and Y are normally distributed.
Since p = 0.042 < 0.05, we reject H0 and conclude that ρis not equal to 0.
n
(e) rs = 1  6 di2 n(n 2  1) =1 – 6(96)/10(100 – 1) = 0.418.
i 1
(f) The rs is smaller in magnitude than r. However, it still suggests a moderate positive
relationship between cholesterol and triglyceride levels.
(g) Test statistic Ts = rs (n  2) /(1  rs2 ) = .418 (10  2) /(1  .4182 ) = 1.30, (for n ≥ 10)
Under H0, the quantity Ts ~ t8. In this case, p = 0.229 > 0.05, we cannot reject H0.
De scri ptive Statistics
Choles terol Level
Tri glyc eride Level
Mean
6.7330
5.9090
St d. Deviat ion
1.56331
3.81842
N
10
10
生統 96-1 HW#11&12
慈濟醫資系 2008/01/09 P.4
Correlations
Spearm an's rho
Choles terol Level
Tri glyceride Level
Choles terol
Level
1.000
.
10
.418
.229
10
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
Tri glyceride
Level
.418
.229
10
1.000
.
10
Correlations
Choles terol Level
Triglyceride Level
Pearson Correlation
Sig. (2-tailed)
Sum of Squares and
Cross-products
Covariance
N
Pearson Correlation
Sig. (2-tailed)
Sum of Squares and
Cross-products
Covariance
N
Choles terol
Level
1
Triglyceride
Level
.650*
.042
21.995
34.902
2.444
10
.650*
.042
3.878
10
1
34.902
131.223
3.878
10
14.580
10
*. Correlation is s ignificant at the 0.05 level (2-tailed).
18.9
(b) ( ̂ , ˆ ) = (10.552, 1.264) → a fitted line ŷ = ˆ  ˆ x = 10.552 + 1.264 x .
The slope implies that each one-week increase in gestational age causes an
infant’s systolic blood pressure to increase by 1.264 mm Hg on average. Although
it does not make sense in this case, the y-intercept of 10.662 is the predicted
systolic blood pressure for an infant whose gestational age is 0.
(c) Test for H0 :β= 0 vs H1 :β≠ 0 , (test for no linear relationship)
test stat t = ˆ seˆ( ˆ ) = 1.264/0.436 = 2.899 ~ t98 , under H0, p = 0.005 < 0.05.
We reject H0 at 0.05 level and conclude that for low birth weight infants, systolic
blood pressure increases in magnitude as gestational age increases.
(d) For x = 31, ŷ = 10.552 + 1.264 x = 10.552 + 1.264(31) = 49.736.
(e) seˆ( yˆ ) = sy|x
1
( x  x )2

n  ( xi  x ) 2
= 11.0
1
4.4521 = 1.434,

100 635.694
for x = 31, x =28.89, ( x  x )2 = 4.4521,
(x  x )
i
2
= 635.694.
Construct a 95% C.I. for ŷ : ( ŷ –t98, .025× seˆ( yˆ ) , ŷ + t98, .025× seˆ( yˆ ) )
= 49.736 ± 1.98(1.434) = (46.91, 52.59).
生統 96-1 HW#11&12
慈濟醫資系 2008/01/09 P.5
Model Summaryb
Model
1
R
.281a
R Square
.079
Adjusted
R Square
.070
Std. Error of
the Estimate
11.000
a. Predictors: (Constant), gestational age
b. Dependent Variable: sys tolic blood press ure
Coeffi cientsa
Model
1
Unstandardized
Coeffic ient s
B
St d. E rror
10.552
12.651
1.264
.436
(Const ant)
gestat ional age
St andardiz ed
Coeffic ient s
Beta
t
Sig.
.406
.005
.834
2.898
.281
95% Confidenc e Interval for B
Lower Bound Upper Bound
-14.553
35.657
.399
2.130
a. Dependent Variable: sy stolic blood pres sure
18.11
(a) The slope implies that for each 1% increase in contraceptive practice, total fertility
rate decreases by 0.062 births per woman. The intercept of 6.83 is the predicted
total fertility rate for a country with 0% prevalence of contraceptive practice.
(b) The two-way scatter plot is displayed below. It appears that total fertility rate
decreases as contraceptive practice increases.
9.00
fe rtility ra te p e r wo m a n
8.00
7.00
6.00
R S q Line a r = 0.22
5.00
0.00
10.00
20.00
30.00
40.00
50.00
c o n tra c e p tio n u s e (% )
(c)
(d) The total fertility rates for the African nations are higher than would be predicted
based on the regression line. ŷ = ˆ  ˆ x = 6.947 – 0.030 x
Coeffi cientsa
Model
1
(Const ant)
contracept ion use (%)
Unstandardized
Coeffic ient s
B
St d. Error
6.947
.261
-.030
.014
a. Dependent Variable: fertility rat e per woman
St andardiz ed
Coeffic ient s
Beta
-.469
t
26.628
-2. 056
Sig.
.000
.058