CmpE 343
Spring 2010
Problem Session#10
Solution Key
Question1: Suppose that a box contains four marbles, θ of which are red and 4 − θ of which are blue.
Consider testing H0 : θ = 2 versus H1 : θ 6= 2 as follows: Draw two marbles, with replacement, and
reject H0 if both marbles are of the same color. Otherwise, do not reject.
(a) What is the probability of a Type I error?
(b) What is the probability of a Type II error if θ = 3?
Solution1:
(a)
Pr(Type I error) = Pr(reject H0 when true)
= Pr(both marbles are the same color when θ = 2)
= Pr(both red when θ = 2) + Pr(both blue when θ = 2)
22 22
1
+
=
=
44 44
2
(b)
Pr(Type II error) = Pr(do not reject H0 when false)
= 1 − Pr(reject H0 when false)
= 1 − Pr(both marbles are the same color when θ = 3)
= 1 − Pr(both red when θ = 3) − Pr(both blue when θ = 3)
3
33 11
−
=
=1−
44 44
8
2
Question2: Let X ∼ N (µX , σX
) and Y ∼ N (µY , σY2 ) with known variances. We sample X1 , . . . , XnX
independently from X, and Y1 , . . . , YnY independently from Y . How can we test H0 : 2µX − 3µY = 5
versus H1 : 2µX − 3µY > 5 with significance level α?
Solution2: We should test the following hypotheses:
H0 : 2µX − 3µY = 5
H1 : 2µX − 3µY > 5 .
2X̄ − 3Ȳ − 5
Z=r 2
4σX 9σY2
+
nX
nY
If Z < zα , we can not reject the null hypothesis.
Question3: A study was instituted to learn how the diets of people changed during winter and summer.
A random group of n people were observed during both seasons and the percentages of each person’s
calories that came from fat were determined. Describe a proper statistical procedure to test the hypothesis that the fat percentage intake is the same for both seasons versus the hypothesis that the fat
percentage intake in winter is larger than that in summer.
Solution3: Proper statistical test is the paired t-test.
Person
1
2
..
.
Winter
w1
w2
..
.
Summer
s1
s2
..
.
Difference
d1 = w 1 − s 1
d2 = w 2 − s 2
..
.
n
wn
sn
dn = w n − s n
v
uP
u n (d − d)2
di
u
i
t
.
We can construct the procedure as follows with d¯ = i=1 and sd = i=1
n
(n − 1)
n
P
H0 : µd = 0
H1 : µd > 0
We know that
if
d
√ follows a t-distribution with υ = n − 1 degrees of freedom. We can not reject H0
sd / n
d
√ ≤ tα,n−1 or reject H0 otherwise.
sd / n
Question4: Suppose we are testing the following hypothesis
H0 : µ1 − µ2 = 0
H1 : µ1 − µ2 > 0
with significance level α where σ12 and σ22 are known. Resources are limited and consequently the total
sample size n1 + n2 = N . How should we allocate the N observations between the two populations to
obtain the most powerful test for the alternative hypothesis µ1 − µ2 = δ?
Solution4:
Power = 1 − β



a−δ 
 where a = zα
r
= 1 − Pr 
Z
<

σ12 σ22 
+
n1 n2



= 1 − Pr 
Z < zα − r
r
We should minimize
δ
s
σ12 σ22
+
n1 n2


σ2 
σ12
+ 2
n1 n2
σ12 σ22
+
in order to get the most powerful test.
n1 n2
σ12
σ22
+
n1 N − n1
σ2
σ22
∂f (n1 )
= − 12 +
=0 ⇒
∂n1
n1 (N − n1 )2
σ2
n2 = N − n1 =
N
σ1 + σ2
f (n1 ) =
n1 =
σ1
σ1 + σ2
N
Question5: A question of medical importance is whether jogging leads to a reduction in one’s pulse
rate. To test this hypothesis, 8 non-jogging volunteers agreed to begin a 1-month jogging program.
After the month their pulse rates were determined and compared with their earlier values. If the data
are as follows, can we conclude that jogging has an effect on the pulse rates with the level of significance
α = 0.05?
Subject
1
Pulse Rate Before 73
Pulse Rate After
70
2
87
84
3
98
90
4
102
110
5
78
71
6
87
80
7
77
69
8
70
74
Solution5:
Subject
1
Difference 3
2
3
3
8
4 5 6 7
-8 7 7 8
8
-4
H0 : µD = µ1 − µ2 = 0
H1 : µD = µ1 − µ2 6= 0
d¯ = 3
sd = 6
3−0 √
T = √ = 2 ≈ 1.4142
6/ 8
T < tα=0.025,υ=7 = 2.3646, there is no evidence that jogging has an effect on the pulse rate.