CHAPTER 8. FUNDAMENTALS OF STATISTICAL ANALYSIS
CHAPTER 8. FUNDAMENTALS OF STATISTICAL ANALYSIS ………………………………………..1
8.1. Introduction ………………………………………………………………………………………………………………1
8.2. Estimation of Parameters …………………………………………………………………………………………….1
8.3. Method of Moments Estimation ………………………..…………………………………………………………3
8.4. Maximum Likelihood Estimation ……………………….……………………………………………………….8
8.5. Sampling Distributions ……………………………………………………………………………………..………..8
The following table provides a summary of the problems with their appropriate sections:
Section Problems
8.1 1
8.2 2 to 8
8.1. Introduction
None
8.2. Estimation of Parameters
Problem 8-1.
The samples are incomplete information. The samples are collected to draw inferences about the
Problem 8-2.
(a) The times of the last five Kentucky Derby winners
Population: the times of all the Kentucky Derby winners
8-2
Problem 8-3.
Problem 8-4.
The archer would hope to hit the bull’s eye with each shot. However, this is unlikely. If the archer
Problem 8-5.
The officer’s radar gun measures speed to some accuracy level. The accuracy has two components
of bias and precision. If the gun is not well-calibrated, it could have a bias by consistently
measuring the speed at a higher level than the actual speed. The bias could also be on the lower
side. The precision is relating to the nearest unit of speed that can be measured by the gun, e.g.,
within a mile per hour precision or within 5 miles per hour precision.
Problem 8-6.
Bias: Due to different strengths and weaknesses of a student, one would expect that the grades for
different reports exams and quizzes would have an amount of bias compared to the final grade that
each student gets. A grader of assignments could also be biased based on factors such as the
legibility of the writing. A grader who is lenient (or very tough) could bias the grades.
8-3
Accuracy: It is the hope of the student and the professor that the grade that the student gets
represents the amount knowledge that the student has gained from the class. Biases or lack of
Problem 8-7.
Method 1: mean = 102.14, and standard deviation = 0.3647
Method 2: mean = 100.60, and standard deviation = 1.5166
Therefore method 2 is less biased, and method 1 is more precise.
Problem 8-8.
8.3. Method of Moments Estimation
Problem 8-9.
The most likely value is 1.
Problem 8-10.
8-4
S =
S2= 0.8111
This is not possible because X cannot take on a value of 0.61.
Problem 8-12.
Largest:
Smallest:
Problem 8-13.
Problem 8-14.
8-5
The sum of these values is 1.
Problem 8-15.
Problem 8-16.
P=xf x dx x x dx x dx
1
32
1
33
1
3
3
26
3
26
³³ ³
§
©
¨·
¹
¸
()
Problem 8-17.
8-6
Problem 8-18.
Problem 8-19.
Problem 8-20.
Two requirements: 1)(
³
b
a
dxxf and 4)(
³
b
a
dxxxf
Problem 8-21.
Problem 8-22.
yo and xo must satisfy the constraint 1)(
³ dxxf
The second equation is based on the mean
6
Solving Eq.A for yo:
Problem 8-23.
8.4. Maximum Likelihood Estimation
Problem 8-24.
8.5. Sampling Distributions
Problem 8-25.
Problem 8-26.
Friction coefficient (X): N(0.005, 0.0002)
Mean friction coefficient (
X
): N(0.005, 0.0002/ 5 ) = N(0.005, 0.0000894)
3500
4000
4500
5000
f(x)
f(xͲmean)
8-9
Problem 8-27.
Friction coefficient (X): N(11, 2)
Problem 8-28.
(a) P(random pump has efficiency of 75% or greater):
P(X>75) = 1 – P(X<75) = 1 – )[(75-72)/3.2] = 1 – )[.9375] = 1 – 0.8258 = 0.1742 = 17.42%
Problem 8-29.
Problem 8-30.
(a) n = 20, find: probability that
x
> 2600 s/cm
Problem 8-31.
n = 15,
P
= 32%,
V
= 2%
Problem 8-32.
The sampling distribution of sample mean is
Problem 8-33.
2
2
28
F
V
S
= 2
2
)002.0(8
F
Problem 8-34.
D
F
^2 V^2 V
0.95 2.73 1.17216EͲ05 0.003423684
Plotting the density function requires differentiating D to obtain the density function, then plotting
V2 versus the density value.
As for the mean, the following plot can be created:
Problem 8-35.
(a) P=10
Problem 8-36.
The solution for this problem is not provided.
DF
^2 V^2 V
0.995 2.603 270.4571648 16.44558
0.14
0.16
0.18
0.2
f(x)
f(xͲmean)
8-12
Problem 8-37.
(a) VF
VF
22
2
2
2
222
319
3313
?
() (/)
nS S SS
Problem 8-38.
Problem 8-39.
The solution for this problem is not provided.