Reject H0 if TS > TV or TS < –TV One Tailed Test Alternative hypothesis having one side.. Decision rule Reject H0 if TS < TV Hypothesis Testing Procedure It is based on sample statist
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CV = Critical Value
SE = Standard Error
∝ = Level of Significance
TS = Test Statistics
TV = Table Value
Two Tailed Test
Alternative hypothesis having two sides
H0: µ = µ0 vs Ha µ≠µ0.
Reject H0 if
TS > TV or TS < –TV
One Tailed Test
Alternative hypothesis having one side
Upper Tail
H0:µ≤µ0 vs Ha: µ > µ0
Decision rule
Reject H0 if TS > TV
Lower Tail
H0:µ≥µ0 vs Ha: µ < µ0
Decision rule
Reject H0 if TS < TV
Hypothesis Testing Procedure
It is based on sample statistics & probability theory
It is used to determine whether a hypothesis is a reasonable statement or not
Steps in Hypothesis Testing
1.State the hypothesis
2.Identify the appropriate test statistic and its probability distribution
3.Specify the significance level
4.State the decision rule
5.Collect the data &
calculate the test statistic
6.Make the statistical decision
7.Make the economic or investment decision
Hypothesis
Statement about
one or more
populations
Null
Hypothesis H0
Tested for
possible
rejection
Always
includes ‘=’
sign
Two Types
Alternative Hypothesis
Ha Hypothesis is accepted when the null hypothesis is rejected
(Source: Wayne W Daniel and James C Terrell, Business Statistics, Basic Concepts and Methodology, Houghton Mifflin, Boston, 1997.)
Statistical Significance vs
Economical Significance
Statistically significant results
may not necessarily be
economically significant
A very large sample size may
result in highly statistically
significant results that may be
quite small in absolute terms
Significance Level ( α )
Probability of making a type I error
Denoted by Greek letter alpha (α )
Used to identify critical values
Two Types of Errors Type I Error
Rejecting a true null hypothesis
Type II Error
Failing to reject a false null hypothesis
Decision Rule
Based on comparison of TS to specified value(s)
It is specific & quantitative
Test Statistics
Hypothesis testing involves two statistics:
TS calculated from sample data
Critical values of TS
Trang 22017, Study Session # 3, Reading # 12
Population Mean
• σ2
known
x z
σ
−
= • Ho:µ ≤ µ0 vs Ha: µ >µ0
Reject H0 if TS > TV
• Ho:µ > µ0 vs Ha: µ <µ0
Reject H0 if TS < –TV
• Ho:µ = µ0 vs Ha: µ ≠µ0
Reject H0 if TS > TV or TS < – TV
• n ≥ 30
• σ2
unknown
= ̅
√
or ∗ = ̅
*(more conservative)
• σ2
unknown
• n<30
x
tn
σ
µ0 1
−
=
Equality of the
Means of Two
Normally
Distributed
Populations based
on Independent
Samples
Unknown variances assumed equal
2 1
2 1 2 1 ) (
1 1
) (
) (
2 2 1
n n s
x x n
n t
−
−
−
=
− +
µ µ
where;
2
) 1 ( ) 1 (
2 1
2 2 2 2 1 1
− +
− +
−
=
n n
s n s n
sP
df = n1+n2 - 2
• Ho:µ1 - µ2 ≤ 0 vs Ha: µ1 -µ2 > 0 Reject H0 if TS > TV
• Ho:µ1 - µ2 > 0 vs Ha: µ1 -µ2 < 0 Reject H0 if TS < -TV
• Ho:µ1 - µ2 = 0 vs Ha: µ1 - µ2 ≠ 0 Reject H0 if TS > TV or TS < – TV
Unequal unknown variances
2
2 2 1
2 1
2 1 2
(
n
s n s
x x t
+
−
−
−
2
2
2
2 2
1
2
1
2 1
2
2
2 2 1
2 1
.
n n s
n n s n
s n s f
d
+
+
=
df = Degree of Freedom
n ≥ 30 = Large Sample n< 30 = Small Sample
n = Sample Size
σ2
= Population Variance
N.Dist = Normally Distributed
N.N.Dist = Non Normally Distributed
Power of a Test
1 – P(type II error)
Probability of correctly rejecting
a false null hypothesis
p- value
The smallest level of significance
at which null hypothesis can be rejected
Reject H0 if p-value < α
Relationship b/w Confidence Intervals &
Hypothesis Tests
Related because of critical value
C.I
[(SS)- (CV)(SE)] ≤parameter ≤[(SS) + (CV)(SE)]
It gives the range within which parameter value
is believed to lie given a level of confidence
Hypothesis Test
-C V≤TS≤+ CV
Range within which we fail to reject null
hypothesis of two tailed test at given level of
significance
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Testing Variance of a N.dist Population
TS
Decision Rule
Reject H0 if TS > TS
Chi-Square Distribution
Asymmetrical
Bounded from below
by zero
Chi-Square values can never be –ve
Testing Equality of Two Variances from N.dist Population TS
; >
Decision Rule
Reject H0 if TS > TV
F- Distribution
Right skewed
Bounded by zero
Parametric Test
Specific to population parameter
Relies on assumptions regarding the distribution of the population
Non-Parametric Test
Do not consider a particular population parameter
Or
Have few assumptions regarding population
Paired Comparisons
Test
TS t(n-1 )=
̅ = 1
S= S
√n
= ∑ ( − ̅)
− 1
Decision Rule
H0: µd ≤µd0 vs Ha: µd > µd0
Reject H0 if TS > TV
H0: µd ≥µd0 vs Ha:µd < µd0
Reject H0 if TS <-TV
H0: µd = µd0 vs Ha: µd ≠µd0
Reject H0 if TS > TV
...̅ = 1< /sup>
S=...
√n
= ∑ ( − ̅)
− 1< /h3>
Decision Rule
H0: µd ≤µd0 vs Ha: