Section 8.7
Exercise 8.51. given by A 90% con dence interval for p1
b (p1 b p2 )
p2 is approximately
1:645
s
(a) If n1 = 800, n2 = 640, x1 = 337, x2 = 374, then
b p1 =
b b b b p 1 q1 p 2 q2 + . n1 n2
x1 337 = = 0:421 n1 800
s
and
Hence, the i
MTH/STA 562
Exercise 7.1. Since the amount of .ll dispensed by a bottling machine is normally distributed with = 1, the sample mean will also be normally distributed, from Theorem 7.1, with mean and variance 2 = 2 =n. Y (a) If n = 16, then P Y 0:3 =
CHAPTER 10
INFERENCE FROM SMALL SAMPLES
10.2 Students t Distribution
From the discussion of the sampling distribution of E in the preceding chapters, a few
points had been discovered:
* When the original sampled population is normal, the statistics
T
CHAPTER 9
LARGESAMPLE TESTS
OF HYPOTHESES
9.1 Testing Hypotheses about Population Parameters
Often statisticians are not interested solely in estimating unknown population parameters,
but in deriving the formulation of rules or procedures that leads to
CHAPTER 7
SAlVIPLING DISTRIBUTION
7.2 Sampling Plans and Experimental Designs
The way a sample is selected is called the sampling plan or experimental design. Simple
random sampling is commonly used sampling plan in which every sample of size n has the
Section 10.7
Exercise 10.58. It is given that n1 = 16, n2 = 20, s2 = 55:7, and s2 = 31:4. 1 2
2 1 2 2 2 1 2 2:
(a) The null and alternative hypotheses for the test are H0 : The test statistic is = versus Ha : 6=
s2 55:7 1 = = 1:774: 2 s2 31:4 With
Section 10.6
Exercise 10.49. interval for 2 is It is given that n = 15 and s2 = 0:3214. A 90% con dence (n
2 =2
1) s2
<
2
<
(n
2 1
1) s2
=2
:
With
= 0:10 and df = n
2 =2
1 = 14, we read from Table 5 that and
2 2 1 =2
=
2 0:05
= 23:6848
Section 10.5
Exercise 10.35. It is given that n = 10, d = 0:3, s2 = 0:16 (sd = 0:4). d (a) The null and alternative hypotheses for the test are H0 : versus Ha : The test statistic is t= d 0:3 0 p = 2:372: pd = sd = n 0:4= 10
1 2 1 2
=0 6= 0
or or
Section 10.3
Exercise 10.1. (a) 2:015 (b) 2:306 (c) 1:330 (d) 1:960 Exercise 10.2. (a) A two-tailed test with = 0:01 and 12 df gives t =2 = t0:005 = 3:055. The rejection region is jtj > 3:055; that is, either t < 3:055 or t > 3:055.
(b) A right-tail
Section 9.6
Exercise 9.43. (a) The hypotheses are H0 : p 1 p2 = 0 versus Ha : p 1 p2 < 0:
(b) This is a left-tailed test. (c) With n1 = 140, n2 = 140, x1 = 74, x2 = 81, the two individual sample b b proportions are p1 = 74=140 and p2 = 81=140, respe
Section 9.5
Exercise 9.31. (a) The hypotheses are H0 : p = 0:4 This is a two-tailed test.
b (b) With n = 1400 and x = 529, the sample proportion is p = 529=1400 = 0:378 and the test statistic is
versus
Ha : p 6= 0:4:
0:378 z= q
0:4
(0:4)(0:6) 14
Section 8.9
Exercise 8.68. The parameter on interest is . Then n z 2 =2 2 : B2 = 0:95 or = 0:05, we have
It is given that = 12:7 and B = 1:6. With 1 z =2 = z0:025 = 1:96. Hence, n or n = 243:
(1:96)2 (12:7)2 = 242:036 (1:6)2
Exercise 8.69.
The pa
Section 8.5
Exercise 8.23. The 90% con dence interval for x 1:645 p . n is given by
(a) If n = 125, x = 0:84, and s2 = 0:086, then p 0:086 = 0:084 0:043 or 0:084 1:645 p 125 (b) If n = 50, x = 21:9, and s2 = 3:44, then p 3:44 21:9 1:645 p = 21:9 0:4
Section 8.4
Exercise 8.3. The margin of error in estimating Margin of error = 1:96 (a) If n = 30 and
2
is given by
p . n
= 0:2, then p = 1:96 n p 0:2 p = 0:160. 30
1:96 (b) If n = 30 and
2
= 0:9, then p = 1:96 n p 0:9 p = 0:339. 30
1:96 (c) If
Section 7.6
Exercise 7.37. The mean and standard deviation of the sampling distribb ution of the sample proportion p are, respectively, given by (a) If n = 100 and p = 0:3, then
b E (p) = 0:3 b E (p) = p
and
b SE (p) =
r
pq : n
and
(b) If n = 4
Section 7.5
Exercise 7.19. The mean and standard deviation of the sampling distribution of the sample mean x are, respectively, given by E (x) = (a) If n = 36, = 10, and
2
and = 9, then
SE (x) = p : n
E (x) = 10 (b) If n = 100, = 4, and
and
2
3
Section 7.2
Exercise 7.1. You can select a simple random of size n = 20 using Table 10 in Appendix I. First choose a starting point and consider the rst three digits in each number. Since the experimental units have already been numbered from 000 to