.
3.10
(i) Set 8 = &, a 2). Then from E e(CLl ciXi) = p , We get p Ci = P f or all p , SO that EL1 i = 1. This is the condition that the C~'S must c satisfy, so that Cy=l iXi is an unbiased estimate of p . c (ii) Vare(U,) = V ~ r e ( C y = ~Xi) = a 2Cy=I
Statistics 131B
Winter Quarter 2010
H W #21 (Chapter 11)
4.1 A coin, with probability 6' of falling heads, is t ossed independently 100
times and 60 heads are observed. At level of significance cr = 0.1:
(i) Use the LR t est in o rder to test the hypothes
Winter Quarter 2010
Statistics 1 31B
HW #20 (Chapter 1 1 ): Answers
3.7 (ii) By Application 2 , t he test for t esthg the hypothesis Ha is d e h e d by
relations ( 26) and ( 27).
Ciii) With Slw=
X i, we have:
c
:
Thus, Ho is rejected when Sloe < 19,767. B
Winter Quarter 2010
Statistics 1 3 1 ~
H W #20 (Chapter 11)

3.7 /ii) In reference to Example 12 in Chapter 1, t he appropriate model to
be used is t he Poisson model; i.e., X P(A).
(ii); The safety level is specified by A 5 200, and this is checked by t
Winter Quarter 2010
S tatistics 1 31B
HW # 1 9 (Chapter 1 1 ): Answers
3.1 (ii) T h e h ypohesis Ho is checked by means of the test defined by relations (19) and (20).
X
C 100x0.05
(iii) 0.02 = Po.m(X > C) = Po.m(=?q% > rnx)
21 @
(el,
= 2.055, or C =
Winter Quarter 2010
Statistics 131B
H W #19 (Chapter 1 1 )
3.1 ( i) In reference to Exanlple S in Chapter 1, t he appropriate model is
t he Binomial iiioclel with Xi = 1 if the ith young adult listens to the
program, and XI = 0 otherwise, where P (X; = 1)
Statistics 1 316
Winter Quarter 2010
H W # I 8 (Chapter 1 1): Answers
2.6 (i) Indeed, f ( x; 0) = je4' x Z~o,l(x), which is of the required form
, with C(6) = 118, Q(6) =
strictly increasing, T(x) = x, a nd
h(x) = 1(0,co)(~).
(ii) The roles of Ho and H A
Winter Quarter 2010
Statistics 131B
HW # 18 (Chapter 1 1)
2.6 L etXbear.v. withp.d.f. f (x;6) = i eZfB,x > 0, 6 E R = (0, oo).
(i) Refer to Definition 1 in order to show that f (.; 6) is of the exponential
type(ii) Use Theorem 2 in order to derive the UM
43
1
Winter Quarter 2010
Statistics 131B
H W # 17 (Chapter 1 1): Answers
2.1 ( i ) W ith x = ( X I , . . ,x 16),we have:
L l(x>=
1
(3&>16
exp

[
x(Xi'I i=1
l6
1~1,
s o that
+
is equivalent to f > C (= (9 log Co 8 )/16). Therefore,the MP test is
given by
Winter Quarter 2010
Statistics 131B
H W #17 (Chapter 11)
2 .13 X I, . . ., X ls are independent r.v.'s:
(i) Construct the MP test of the hypothesis Ho: the conlnlon distribution of the Xi's is N (079) against the alternative H A: the common
distribution o
Statistics 131B
Winter Quarter 2010
H W # 16 (Chapter 11)
1.1$ the following examples, indicate which statements constitute a simple
I
and which a composite hypothesis:
(i) X i s a r . ~wh0sep.d.f. f isgiven by f (x) = 2e&, x > 0.
.
(ii) When tossing a c
Statistics 131B
Winter Quarter 2010
H W # 16 (Chapter 11)
1.1$ the following examples, indicate which statements constitute a simple
I
and which a composite hypothesis:
(i) X i s a r . ~wh0sep.d.f. f isgiven by f (x) = 2e&, x > 0.
.
(ii) When tossing a c
Statistics 131B
Winter Quarter 2010
H W #2 1 (Chapter 11): Answers
4.1
(i) W i t f i x = (XI,. .,XI0
01,
(i)
s,
With ). =
we have 2log). 2 X f and Ho is rejected when
=
2log > C , where C 2 Xf;O,l = 2.706. Of c o m e , L(&) = L(OOIX)
Oi(1 ~ ~)lm', l r
Statistics 1 3 1 ' ~
Winter Quarter 2010
HW # 22 (Chapter 11)
4.6 The diameters of certain cylindrical items produced by a mac11111c~
a
r.v.'s distributed as N ( p , 0.01). A sample of size 16 is taken and it is I rrrlC$
t hat 5 = 2.48 inches. If t he des
Winter Quarter 2010
Statistics 1 3 1 ~
H W #22 (Chapter 11): Answers
4.6 The hypothesis of interest here is f io: p = 2.5 (inches), and it is reasonable
to take the alternative HA: # 2.5 by adopting the position that the items
p
cannot be used, if their d
Statistics 1 31B
W inter Quarter 2010
D iscussion SessionTuesday,
Problem
#d
~ a n u a 2 6,20 10 r~
Set
r ./
3.10 t X I, . . . ,X nbe i.i.d. r.v.'s with mean p and variance 02,oth unknown. b %en f a r a ny known constants cl , . . , c, consider the line
Statistics 1 3 1 ~,
I
Winter Quarter 2010
H W # 25 (Chapter 12): Answers
2 7 7'he hypothesis to be tested here is
formula (15) yields:
I

21" 100OO)*
s+( I
h,
+=~
(94 l00y
10
0
.
= 1/6, = 1,. , 6. Then
i
I
+
1
4:ie
p
(103

102
0)
10
0
.

10
0
10
0
1
I
Statistics 131B1
Winter Quarter 2010
H W # 25 (Chapter 12)
4
2 7 A die i cast independently 600times, a ndthe numbers 1through 6 appear
.
s
with the frequencies recorded beIow.
I
Use the appropriate x 2 goodnessoffit test to test fairness for the die
Statistics 1 3 i ~,
W inter Quarter 2010
H W # 24 (Chapter 12): Answers
1.1 For k = 4, the loglikelihood test s t a m c 2 log A in Example 1 becomes
here:
 log PAO ZB log PBO x u log pm +
+
+
 nlogn],
Q
log
and the hypothesis HOis rejected when 2 log
1.1 (i) In reference to Example 18 in Chapter 1, the appropriate probability
model is the Multinomial distribution with parameters n and p, p ~ ,
hAB, o, where p~ through p o a re the probabilities that an individual,
p
chosen at random from among t he n
4.9 (i) The appropriate LR test is given by the relations (75) and (76). Since
m = 9, n = 1 0anda = 0.05, wehave Fml,nlp12F8,g~o.025= 4.1020.
=
Also, for X F,l,nl,
we have P (X > C) = 0.975 = P(% <
=
P(Y <
where Y
Fnl,l
= F9,8,SO that
= 4.3572, and
C
Winter Quarter 2010
Statistics 131B
MW #23 (Chapter 1 1)
4.9 (i) Let Xi, i = 1, . . . , 9 and Yjl j = 1, . . ., 10 be independent r.v.'s from
the distributions N (pl, a;) a nd N (p2, cri), respectively. Suppose that
the observed values of the sample s.d.'
Statistics 131B
Winter Quarter 2010
H W # 1 5 (Chapter 10): Answers
)
l
1.11 (i) F orx 2 8 , F (x; Q = /= ae('@)lBdl = /fde('e)lB
1  e(3e)1~,and 1  F(q&J e('e)/B.
=
= e(te)lfi& =
(ii) By relation (28) in Chapter 6:
1.12 (i) P ( a i T ~ b ) =
Statistics 1 318
Winter Quarter 2010
,
H W # 15 (Chapter 10)
1.11 Consider the p.d.f. f ( x ;a , B ) = $ e(Z")18, 2 a , a E 3,B > 0 (see Exx
e rcise 111 in Chapter 9), and suppose that B is known and a is unknown,
.
a hd denote it by 8 . Thus, we have h
Winter Quarter 2010
Statistics 131B
HW #7 (Chapter 9): Answers
3.1 According to the hint, E e h ( X ) =
(for all 8 ) is equivalent to:
8
S et  = 1 , s o that 8 = L , 8 =  and 8 (1  8 ) n =
118
lit
Then the last equation above becomes: +:im+, x k o g (
Winter Quarter 2010
Statistics 131B
H W #7 (Chapter 9)
3.1 ,IfX is a r.v. distributed as B(n, B), B E R = (0, I ), show that there is n o
unbiased estimate of I /@.
Hint: If h (X) were such an estbnate, then E oh(X) = $ f or
8(~
(0, 1). Write out the expe