Download H 0

Lecture 5: Section B
Class Web page URL:
http://www.econ.uiuc.edu/ECON173
Data used in some examples can be found in:
http://www.econ.uiuc.edu/ECON173/hmodata.xls
http://www.econ.uiuc.edu/ECON173/hmodata_ans.xls
1
Lecture 5: Today’s Topics

Recap: Confidence Interval and sample size
 Hypothesis testing Methodology
 Decision Making: Type I and II Errors
 Test of Mean with known variance
 P-value Approach
 Test of Mean with unknown variance
2
Decision Making and
Consequences
STATES OF NATURE
A
C
T
I
O
Do Not
Reject H0
Reject H0
H0 True
Correct
Confidence=1-a
Type I Error
P(Type I)=a
H0 False
Type II Error
P( Type II)=b
Correct
Power=1-b
N
S
3
a & b Have an
Inverse Relationship
Reduce probability of one error
and the other one goes up.
b
a
4
To buy a mp3 player
A
C
T
I
O
N
STATES OF NATURE
Napster around
Confidence level
Buy mp3
of the test
player
=1-0.2=0.8
P(Type I
Don’t buy
mp3 player error)=0.2
Napster dead
P( Type II
error)=0.1
Power of the test
=1-0.1=0.9
S
5
Hypothesis Testing Process
Assume the
population
mean age is 50.
(Null Hypothesis)
Is X = 20   = 50?
Population
The Sample
Mean Is 20
No, not likely!
REJECT
Null Hypothesis
Sample
6
Definitions-I

Null Hypothesis (H0): The hypothesis that depicts
the traditional belief or the conventional wisdom
and is maintained unless there is sufficient
evidence to prove otherwise.
 Alternative Hypothesis (H1): The hypothesis
which serves as a plausible alternative to replace
the null hypothesis given there is sufficient
evidence against the null hypothesis.
7
Definitions-II

Type I Error: The error which occurs when you
reject H0 given that it is indeed true.
 Type II Error: The error which occurs when you
do not reject H0 given that it is indeed false.
 Level of Significance (a) : The maximum
probability of committing a Type I Error.
Sometimes (1-a) is called confidence coefficient.
 Power (1-b) : The probability of correctly
rejecting the null hypothesis when it is really false.
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Z test of hypothesis for Mean
(test for , s known, critical
value approach)
Critical
values of z
Area=0.1
0.90
0
Rejection
Region
Rejection
Region
Region of
Acceptance
At 10% level, reject H0
if z is in the Rejection
region. Do not reject if z
is in the Acceptance
region at 10% level. 9
Level of Significance, a
and the Rejection Region
a
H0:   3
Lower one-tailed
H1:  < 3
Rejection
Regions
0
H0:   3
Upper one-tailed
H1:  > 3
0
H0:  = 3
H1:   3
Critical
Value(s)
a
a/2
Two-tailed
0
10
Z test of hypothesis for Mean
(test for , s known,
p-value approach)
Area=p-value
0
Rejection
Region
Rejection
Region
Region of
Acceptance
Test statistic z
At 10% level, reject H0
if p-value<0.1. Do not
reject if p-value0.1.
11
Definitions-III

test statistic: The measured value of the
statistic which is used to test a hypothesis.
 critical value: The tabulated value of the test
statistics, beyond which we reject the null
hypothesis.
 p-value: The smallest level of significance
at which the null hypothesis is rejected.
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Steps for Hypothesis Testing-I

Step 1: Setup the null and alternative
hypothesis. e.g. H0:=20 vs. H1:20
 Step 2: Collect data and decide on a. e.g.
Data on a sample of Doritos and a=0.05.
 Step 3: Calculate summary sample statistics.
e.g. Calculate and s.
 Step 4: Calculate the test statistic z (or t). e.g.
13
Steps for Hypothesis Testing-II

Step 5: Find out the distribution of the test
statistics under H0. e.g.
follows a standard
normal distribution if H0 is true.
 Step 6: Obtain the Rejection region using the pvalue or otherwise. e.g. reject if p-value<a or teststatistic is z<lower critical value (or z>upper
critical value or either)
 Step 7: Make your decision of whether to accept or
reject H0. e.g. Reject the null hypothesis that each
bag of Doritos contain 20oz of chips.
 Step 8: Draw your conclusion. e.g. On an average
the weight of each bag of Doritos is different from
20 oz.
14
One-Tail Z Test for Mean
(s Known)

Assumptions
–
–
–

Population Is Normally Distributed
If Not Normal, use large samples
Null Hypothesis Has  or  Sign Only
Z Test Statistic:
z=
x  x
sx
=
x
s
n
15
Rejection Region
H0:   0
H1:  < 0
H0:   0
H1:  > 0
Reject H0
Reject H 0
a
a
0
Must Be Significantly
Below  = 0
Z
Z
0
Small values don’t contradict H0
Don’t Reject H0!
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Example: One Tail Test
Does
an average box of
cereal contain more than
368 grams of cereal? A
random sample
of 25 boxes
_
showed X = 372.5. The
company has specified s to
be 15 grams. Test at the
a=0.05 level.
368 gm.
H0:   368
H1:  > 368
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Example Solution: One Tail
H0:   368
H1:  > 368
Test Statistic:
a
= 0.025
n = 25
Critical Value: 1.645
Reject
.05
0 1.645 Z
Z=
X 
s
= 1.50
n
Decision:
Do Not Reject at a = .05
Conclusion:
No Evidence True Mean
Is More than 368
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p Value Solution
p Value is P(Z  1.50) = 0.0668
Use the
alternative
hypothesis
to find the
direction of
the test.
p Value
.0668
1.0000
- .9332
.0668
.9332
0 1.50
From Z Table:
Lookup 1.50
Z
Z Value of Sample
Statistic
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t-Test: s Unknown

Assumptions
– Population is normally distributed
– If not normal, only slightly skewed & a large
sample taken

Parametric test procedure

t test statistic
X 
t=
S
n
20
Example: One Tail t-Test
Does an average box of cereal
contain more than 368 grams
of cereal? A random sample of
36 boxes showed X = 372.5,
and s = 15. Test at the a=0.01
level.
s is not given,
368 gm.
H0:   368
H1:  > 368
21
Example Solution: One Tail
H0:   368
H1:  > 368
Test Statistic:
X  
372 . 5  368
t =
=
= 1 . 80
S
15
n
36
a
= 0.01
n = 36, df = 35
Critical Value: 2.4377
Reject
.01
0 2.4377 Z
Decision:
Do Not Reject at a = .01
Conclusion:
No Evidence that True
Mean Is More than 36822