11-1
CHAPTER 11. CONFIDENCE INTERVALS AND SAMPLE SIZE
DETERMINATION
CHAPTER 11. CONFIDENCE INTERVALS AND SAMPLE SIZE DETERMINATION ………….1
11.1. Introduction ……………………………………………………………………………………………..……………..1
11.2. General Procedure ……………………………………………………………………………………………………1
The following table provides a summary of the problems with their appropriate sections:
Section Problems
11.1
11.2 1 to 7
11.1. Introduction
None
11.2. General Procedure
Problem 11-1.
A confidence interval defines the range within which the true parameter value is expected to lie
Problem 11-2.
The three elements of a confidence interval include:
– Kest, which is the estimated value of the statistic K
11-2
Problem 11-3.
Hypothesis test Confidence interval
1. State the hypothesis 1. State the hypothesis
2. Identify the theorem that describes the test
statistic
2. Identify the theorem
Problem 11-4.
Similarities: (a) use of same theorem; (b) based on the same level of significance; (c) one or two
Problem 11-5.
Kest +- FdD
Problem 11-6.
In order to define a confidence interval, a theorem must specify sampling distribution and
Problem 11-7.
The factors are:
sample mean ( x); population or sample standard deviation ( sor
V
); the sample size (n); the
confidence coefficient ( )
J
; one sided vs. two-sided.
11.3. Confidence Intervals on Sample Statistics
Problem 11-8.
995.0;005.0;4.13;10;3
J
D
V
xn
11-3
Problem 11-9.
n = 20;
x
= 27,500
(a) V = 3500
Problem 11-10.
96.1);%(4147.0%18.4 025.0
zneedednotsx
Problem 11-11.
CIsidedtwoXnS 9.0;10.0;35.49;6;1309.1
JD
Problem 11-12.
CIsidedtwoXnS 95.0;05.0;3733.0;6;0625.0
JD
Problem 11-13.
CIsidedtwoXnS 95.0;05.0;286.54;7;2606.8
JD
11-4
Since the mean is not in the interval, they cannot assume that the sample is from the same
population.
Problem 11-14.
CIsidedtwoXn 95.0;05.0;258.0;5;1
JDV
Problem 11-15.
9104.080.2
X
nsX
alpha t 2.8+(t(0.4)/10^0.5)
0.995 -3.250 2.389
Problem 11-16.
nx S 25 329 6 51 10%;.;.,D
Problem 11-17.
nx S 25 575 60;;
(a) Q = 24
Problem 11-18.
nx S 5 3 2 01940;.;.
(a) D/2 = 0.025 Q = 4
11-5
Problem 11-19.
717.0;205.0;5 SXn
Problem 11-20.
D/2 = .025 Q = n1 = 4
Problem 11-21.
10.0
2
4147.0;06.1;4
FX
s
Problem 11-22.
22
31;)(25.6772;29.82
X
nmilssmilss
11-6
Problem 11-23.
The upper tail one-sided test is chosen,
oA
oo
H
H
PP
P
P
!
:
:
11.4. Sample Size Determination
Problem 11-24.
V = 25; D = 0.04; V/2 = .02
Problem 11-25.
error = 3.0; V = 4.7; D= 5%; D/2 = .025
Problem 11-26.
V = 70
(a) D = 10%; D/2 = 5%; ZD/2 = 1.645
Problem 11-27.
)3(/05.02/%90
2
2/
2
2/
?
tHntn
DD
DJ
Problem 11-28.
)(778.1)75.0/(/005.02/99.0
2
2/
2
2/
2
2/
ttHntn
DDD
DJ
Problem 11-29.
error = 3; S = 4.7; = D = 5%; D/2 = 0.025
Since V is unknown, use t statistic
11.5. Relationship Between Decision Parameters and Types I and II
Errors
Problem 11-30.
A sample of size N = 4 from a population with known standard deviation (
V
) of 2, has a sample
mean of 10. For one-sided lower
D
= 0.05, Type I error probability is 0.05 and Type II error
probability is provided by the following figure:
Problem 11-31.
A sample of size N = 4 from a population with known standard deviation (
V
) of 2, has a sample
mean of 10. For one-sided lower
D
= 0.01, Type I error probability is 0.01 and Type II error
probability is provided by the following figure:
0.7
0.8
0.9
1
0.7
0.8
0.9
1