AMA1006 Basic Statistics
Lecture 10
Xin GUO
Email: x.guo@polyu.edu.hk
Oce: TU825
Telephone: 3400 3751
Oce Hours: Friday 14:00-16:00
1 / 29
Review
Sampling Distribution of X
mean
standard error
Distribution
Normal Population Other Population
N(, 2 )
with ,
AMA1006 Basic Statistics
Lecture 10
Xin GUO
Email: x.guo@polyu.edu.hk
Oce: TU825
Telephone: 3400 3751
Oce Hours: Friday 14:00-16:00
1 / 29
Review
Sampling Distribution of X
mean
standard error
Distribution
Normal Population Other Population
N(, 2 )
with ,
c. If P (A B) = 0.1, what is P (A B)?
(25). Suppose A and B are two events with P (A B) = 0.8 and P (A B) = 0.2.
a. Find the possible range of P (A).
b. If P (A) = 0.6, what is P (B)?
(26). Suppose A and B are two events with P (A) = 0.6 and P (B) = 0.7.
AMA 1006 Exercise 3
1. Dene suitable populations from which the following samples are selected:
(a) Persons in 200 homes are called by telephone in the city of Richmond and asked
to name the candidate that they favor for election to the school board.
(b)
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TM 2N Bcrvonu‘aliﬂal 0.7).
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AMA1006 BASIC STATISTICS, SUPPLEMENTARY PROBLEMS
Part 3, Sampling Distribution and Estimation
Note: in this set of problems the words nd, estimate, etc., sometimes mean approximate.
We hope students to decide the trade-o between precision and computation
AMA1006 Basic Statistics
Lecture 9
Xin GUO
Email: x.guo@polyu.edu.hk
Oce: TU825
Telephone: 3400 3751
Oce Hours: Friday 14:00-16:00
1 / 28
Review
Uniform distribution: X Uniform(a, b),
1
Density function: f(x) = ba for a < x < b.
for any a x1 < x2 b, P(x1
AMA1006 Basic Statistics
Lecture 9
Xin GUO
Email: x.guo@polyu.edu.hk
Oce: TU825
Telephone: 3400 3751
Oce Hours: Friday 14:00-16:00
1 / 28
Review
Uniform distribution: X Uniform(a, b),
1
Density function: f(x) = ba for a < x < b.
for any a x1 < x2 b, P(x1
Q1. Assume that X Binomial(300, 0.3). Find the following probabilities.
(a). P (27 X < 85)
(b). P (X = 88)
(c). P (X > 100)
(d). P (X 100)
Solutions. We have 300 0.3 = 90 > 5, 300 (1 0.3) = 210 > 5, and 300 0.3 0.7 = 63 so
approximately X N (90, 63). Let
Topic
T i 2
Probability Distribution
y
Section 1 Discrete Random Variable
Section 1.1 Probability Distribution
Definitions:
A random variable is a variable whose value is
determined by the outcome of a random experiment.
A random variable that assumes cou
AMA 1006 - Assignment 2
1. Let X be a random variable with the following probability distribution:
x
-2
3
5
f (x)
0.3
0.2
0.5
Find the standard deviation of X.
2. The random variable X, representing the number of errors per 100 lines of software
code, has
Q2. Let X N(2, 102 ). Find the constants a, b, and c, such that P (X > a) = 0.3, P (X
b) = 0.6, and P (c X 0) = 0.2.
Solutions. We let Z =
X2
10
to have Z N(0, 1).
(1). 0.3 = P (X > a) = P Z >
(2). 0.6 = P (X b) = P Z
b2
= 0.25, so b = 4.5.
10
a2
10
b2
Table of the Standardized Normal Distribution
P
The table gives the probability
P = Pr(Z > z )
0
where Z ~ N(0,1).
z
z
0.0
0.1
0.2
0.3
0.4
.00
0.5000
0.4602
0.4207
0.3821
0.3446
.01
0.4960
0.4562
0.4168
0.3783
0.3409
.02
0.4920
0.4522
0.4129
0.3745
0.3372
Topic
T i 3
Sampling Distribution and
p g
Estimation
Section 1 Estimation
Section 1.1 Point Estimate
Statistical inference enables us to make judgments about a
population on the basis of sample information
information.
The mean, standard deviation, and pr
Q1. Assume that X Binomial(300, 0.3). Find the following
probabilities.
(a). P (27 X < 85)
(b). P (X = 88)
(c). P (X > 100)
(d). P (X 100)
1
Q2. Let X N (2, 102). Find the constants a, b, and c, such that
P (X > a) = 0.3, P (X b) = 0.6, and P (c X 0) = 0.
Topic 1
Probability
Section 1 Experiments and Sample Space
Definitions:
An experiment is defined to be any process which
generates well defined outcomes. By this we mean that on
any single repetition of the experiment one and only one
of the possible expe
Q1. Suppose that X has a normal distribution such that P (X < 116) = 0.2 and P (X <
328) = 0.9. Determine the mean and variance of X.
Solution. Assume X N(, 2 ) and let Z =
0.2 = P (X < 116) = P
We check the table to give
0.9 = P
so
328
116
Z<
X
N(0, 1).
AMA 1006 Exercise 2
1. Determine the value c so that each of the following functions can serve as a probability
distribution of the discrete random variable X:
(a) f (x) = c(x2 + 4) for x = 0, 1, 2, 3;
2
3
(b) f (x) = c
for x = 0, 1, 2.
x 3x
2. The probab
AMA1006, Basic Statistics, Assignment 1, Model Solutions. Feb 23, 2015
Q1. Solution.
The relative frequencies of A = 3/50 = 0.06
The relative frequencies of B = 4/50 = 0.08
The relative frequencies of A B = 6/50 = 0.12
The relative frequencies of A B
[Question]. How many 5-digit numbers can be formed from the integers 1, 2, 3, .,
9 if no digit can appear more than twice?
[Solution]. We divide all the 5-digit numbers satisfying the requirement, into three
groups.
Group 1. No digit appears more than onc
Q1. Suppose that X has a normal distribution such that P (X <
116) = 0.2 and P (X < 328) = 0.9. Determine the mean and
variance of X.
1
Q2. Suppose X N (2, 32). Find E[3X] and Var(3X).
2
Q3. Suppose X N (2, 32). Find P (|X 1| < 2).
3
Q4. Let X Binomial(10
Q1. We keep rolling two dice together until obtaining at least one dice at 6. Find the
probability that we roll exactly n rounds.
Solution. Let X be the number of rounds we roll to get at least one dice at 6. Since for each trial,
the probability that at
Q1. We keep rolling two dice together until obtaining at least one
dice at 6. Find the probability that we roll exactly n rounds.
1
Q2. A drunk man takes a ring of n keys, where only one can open
the door of his home. Since drunk, he has no idea about the