AAEC 4302
ADVANCED
STATISTICAL METHODS IN
AGRICULTURAL RESEARCH
Chapter 12:
Hypothesis Testing
1
Statistical Hypothesis Testing
• Two contradictory hypotheses:
– Null hypothesis - H0
– Alternative hypothesis – H1
• Three sets of hypothesis:
H0 :B j  B j 0
H1 : B j  B j 0
H0 : B j  B j
H1 : B j  B j 0
0
H0 : B j  B j
H1 :
0
0
Bj  Bj
2
Statistical Hypothesis Testing
• Basic significance Test:
H0 : B j  0
H1 : B j  0
Decision rule:
ˆ*
ˆ c
Reject H0 if B
 B
j
j
Reject H0 Do not reject H0 Reject H0
                    
c
ˆ
 Bj
0
Bˆ j
ˆB c
j
3
Statistical Hypothesis Testing
2 types of mistakes:
H0 is true
H0 is false
(H1 is false)
(H1 is true)
_________________________________________
Reject H0
Error –Type I
Correct Decision
__________________________________________
Do not
Reject H0
Correct Decision Error – Type II
_________________________________________________
4
Statistical Hypothesis Testing
Linear transformation that yields a random variable Z
that has a normal distribution ( µ=0, σ=1)
ˆ j   j
Z 
 ( ˆ j )
• Critical value Zc is determined from Pr(|Z|≥ Zc ) = ά
tj

Bˆ

 Bj 
,  j  0,...k 
ˆ
S B j 
j
5
Statistical Hypothesis Testing
• How to conduct the t-test:
1)
2)
3)
4)
5)
State the hypotheses
Choose the level of significance α
Construct the decision rule
Determine the value of the test statistics t*
State and interpret the conclusion of the test
6
Statistical Hypothesis Testing
Example:
Yi = B0 + B1X1 + B2X2 + Ui
^
^
^
Ŷi = B0 + B1X1 + B2X2
Ŷi = 474.05 + 1.46X1 +26.32X2
Where:
Yi = Cotton Yields (lbs/ac)
X1 = Phosphorous Fertilizer (lbs/ac)
X2 = Irrigation Water (in/ac)
7
Interpreting Summary Output from Excel
Intercept
P
W
Bˆ 0 
Bˆ1 
Bˆ 2 
Coefficients Standard Error
t Stat
474.0476233
43.51108281 10.89487075
1.457121257
1.172416643 1.242835698
26.31733728
4.891860687 5.379821498
P-value
4.477E-20
0.21614834
3.32369E-07
Bˆ1  B1 1.457 0
t* 

 1.243
SE1
1.172
Total number of observations 134
8
Some General Remarks
• A “rule of thumb” is that:
|tj*| ≥ 2
 βj is statistically different from zero, at
least at the 95% level of statistical
certainty
9
Some General Remarks
• One-tail test vs. two-tail test
Advantage
• If you properly justify that Xj has only a
positive (negative) effect on the dependent
variable Yi, then the one-tail test will help you
reject the null hypothesis.
• Under a one-tail test, the critical t-value is
smaller than the critical t-value under a two-tail
test.
10
Some General Remarks
• One-tail test vs. two-tail test
Disadvantage
• If you decide that Xj has only a positive
effect on Y, than you cannot change your
decision after running the regression.
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