CHAPTER 7
Sampling
Distributions
and CLT
Sampling
Distribution of
Sample
Mean
BOOK
pp. 353 356
pp. 370 - 373
Theorem 7.1
Theorem 7.4
What if 2 is unknown?
Assumption: population having
normal distribution
If X is the mean of a random sample of size n
tak
SOME FORMULAE
N
n
s2
(yi y) 2
2
i 1
n -1
(y
i 1
i
) 2
N
n
p( y) p y (1 p) ( n y ) for y 0,1,2,., n E (Y ) np, var(Y ) np(1 p), 0 p 1, m(t ) 1 p e t 1
y
p( y) p(1 p) ( y1) for y 1,2,. E(Y ) 1/ p, var(Y ) (1 p) / p 2 , 0 p 1, m(t ) pet / 1 (1 p)et
p
EC 233 - PS 5
1) Finding m.g.f of a distribution:
a) (Ex. 3.145) If Y has a binomial distribution with n trials and probability of success p, find the
m.g.f for Y
b) (Ex. 3.146) Find the mean and the variance of Y by using m.g.f
2) Finding m.g.f of a dist
PS 4 EC 233
1) A student takes an exam that consists of 10 multiple choice questions. Each question has 5
possible answers. Suppose the student knows nothing about the subject and just guesses the
answer on each question. What is the probability that this
EC 233 - PS 8 Questions /Answers
6. 56 The number of planes arriving per day at a small private airport is a random variable having a
Poisson distribution with = 28.8. What is the probability that the time between two such arrivals is
at least 1 hour?
6.5
EC 233 PS 6 questions
1) You are given the following pdf.
f(x): 1/3 for 0<x<1
1/3 for 2<x<4
0 elsewhere
a) Plot the graph of this pdf
b) Find the cumulative distribution function and plot its graph.
Q: The weekly gasoline demand at a filling station is a random variable normally distributed with
mean 3000 lt/week and st. deviation 200 lt. At the beginning of a certain week, the storage tank of
the filling station is empty.How much gasoline should the
EC 233 - PS 3 Chapter 3
(Probability Distribution of Discrete R.Vs; Mean and Variance; Binomial Distribution)
0.
a)
b)
c)
d)
Tell whether each of the following is a discrete or a continuous random variable.
The number of beers sold at a bar during a parti
EC 233 - PS 2
1) (Textbook 2.66) Suppose that the number of distributors is M = 10 and that there are n =7
orders to be placed.
a) What is the probability that all of the orders go to different distributors?
b) Distributor I gets exactly two orders and Di
Chapter 1.2
Characterizing a Set of Measurements:
Numerical Methods
1
There are some measures used to summarize the data
contained in a sample. Well study their properties in detail
later on and see more of them. For now lets see the most
basic ones.
Nume
Boazii University
Fall 2013
EC 233
Mathematical
Statistics 1
Instructor: Gzin Glsn Akn
Office: NB 203
Office hours: FF 23
Check out the class webpage.
You can download the syllabus
and hence all the information
presented here from there. Go
to www.econ.bo
Writing an Empirical Research Report, and
Sources of Economic Data
Chapter 17
Prepared by Vera Tabakova, East Carolina University
17.1.1 Choosing a Topic
Select an area of interest and identify a problem you wish to work on.
Find suitable and readily avai
Sampling Distributions
A sampling distribution is created by, as the name suggests,
sampling.
The method we will employ on the rules of probability and
the laws of expected value and variance to derive the
sampling distribution.
For example, consider the
Inferential Statistics
Making statements about a population by
examining sample results
Sample statistics
(known)
Population parameters
Inference
Sample
(unknown, but can
be estimated from
sample evidence)
Population
Why Sample?
The cost is less
The da
Common
Large/Small Sample
Tests
1/55
Test of Hypothesis
for the Mean ( Known)
Convert sample result ( x ) to a z value
Hypothesis
Tests for
Known
Unknown
Consider the test
H0 : = 0
The decision rule is:
x 0
Reject H0 if z =
> z
(Assume the population i
EC 233
Fall 2016
Team member names and IDs:
1.
Last Name
First Name
ID
KAHRAMAN
Simay
2015300237
2.
3.
Assignment I for EC 233
(due by Monday, October 10 2016 at 23:55)
This assignment is to be completed by teams of one, two or three students.
You will
Some useful formulae for Midterm I
2 = E(X ) 2 = (x ) 2 P(x) or
= E(x) = x P(x)
x
2 = E(X 2 ) - 2
x
N
x
x + x2 + xN
= 1
=
N
N
i =1
N
=
2
n
i
(x
i =1
x=
x
i =1
n
) 2
i
i
=
n
S2 =
N
E[g(X)] = g(x)P(x)
(x
i =1
i
x1 + x 2 + + x n
n
x) 2
n -1
E(bX) = b
Important Terms
EC 233.01/02
Lecture Notes 5
Probability
y (Chapter
p 2)
Random Experiment a process leading to an
uncertain outcome
Basic Outcome a possible outcome of a
random experiment
Sample Space the collection of all possible
outcomes of a random e
BayesTheorem
BayesTheoremisusedtorevisepreviously
y
p
y
calculatedprobabilitiesbasedonnewinformation.
EC 233
233.01/02
01/02
Lecture Notes 7
DevelopedbyThomasBayesinthe18th Century.
Bayes
y Theorem (Chapter
p 2)
Itisanextensionofconditionalprobability.
Introduction to
Probability Distributions
Random Variable
Represents a possible numerical value from
a random experiment
EC 233.01/02
Random
Variables
Lecture Notes 8
(Chapter 3)
Ch. 3
Discrete
Random Variable
Continuous
Random Variable
Ch. 4
1
2
Discre
PS on Extra Session Topics: Questions and Answers
Note: There is a correction in the below table. Check with the answer in the problem session
Ex. 5.3 is as follows: