1
ECON1320
Quantitative Economic and
Business Analysis B
LECTURE 2
Chi Square tests
Chapter 12.1 – 12.2
2
Topics
Use the Chisquare (
χ
2
) statistic to test
1.
whether a given variable follows a
hypothesized population distribution (
χ
2
GoodnessofFit Test)
2.
hypotheses on population proportions as an
alternative to the
z
test
3.
whether two or more categorical variables are
independent (
χ
2
Test of Independence)
3
ChiSquare distribution
r
Statistical experiment: select a random sample of size
n
from a normal population, having a SD = σ. We find
that the sample SD =
S
.
r
Compute a statistic, called
chisquare
, using the
following equation:
r
If this experiment is repeated an infinite number of
times, we could obtain a sampling distribution for the
chisquare statistic.
2
2
2
*
)
1
(
σ
χ
S
n

=
4
ChiSquare distribution
r
always positive – first quadrant
r
there is a family of
χ
2
distributions
r
a particular member is specified by the
degrees of freedom
r
right skewed for
low d.f.
r
more
symmetrical as
d.f. increase
χ
2
0
df = 1
5
5
Chisquare statistic
r
used for qualitative (categorical) data
r
uses count data – frequency (not
proportion or %) of success
r
contingency table (shows success/failure
for two or more groups)
r
χ
2
statistic compares observed
frequency (f
o
) with expected frequency
(f
e
) in each cell
6
Chisquare statistic
r
the test statistic is given by
χ
2
calc
=
r
d.f. = km1,
where
k
= the number of
categories,
m
= the number of parameters
estimated from the sample data (used to
determine expected frequencies)
NB: if expected distribution is given, m=0, df=k1
r
all expected frequencies must be
≥
5, if not,
cells must be combined to have more than 5
∑

cells
all
e
e
o
f
f
f
2
)
(
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
7
Topic1 :ChiSquare GoodnessofFit Test
r
is used to determine whether a given variable
follows a hypothesised population distribution
(i.e. normal, uniform, binomial or multinomial
distribution)
r
determines how well the theoretical distribution
fits the observed data (how good is the fit?)
r
compares an
observed
frequency distribution
with a
theoretically expected
distribution.
8
The fivestep test
r
Step 1: state hypotheses
H
0
: The variable follows the … distribution
H
1
: The variable does not follow the …
distribution
Or
H
0
: No change to the distribution
H
1
: The distribution has been changed
9
The fivestep test
r
Step 2: decision rule: reject H
0
if
χ
2
cal
>
χ
2
α
(k1)
Non Rejection
region
Rejection
region
χ
2
α
(km1)
α
10
The fivestep test
r
Step 3: compute test statistic &
compare with the decision rule
r
Step 4: make a decision
r
Step 5: state your conclusion
2
2
all cells
(
)
o
e
calc
e
f
f
f
χ

=
∑
11
Example 1
r
A wholesale distributor buys goblets
from a supplier. The supplier claims
that they are producing goblets at a
10% defective rate. On every
delivery, the distribution company
check randomly 10 goblets and
record this statistic.
r
This is the end of the preview.
Sign up
to
access the rest of the document.
 Three '08
 JOHN
 ChiSquare Test, Statistical hypothesis testing, Chisquare distribution, Pearson's chisquare test

Click to edit the document details