PEP-PMMA Training Session
Statistical inference
Lima, Peru
Abdelkrim Araar / Jean-Yves Duclos
9-10 June 2007
Why statistical inference?
• Distributive estimates obtained from surveys are
not exact population values.
• The estimates normally follow a known
asymptotic distribution. The parameters of that
distribution can be estimated using sample
information (including sampling design).
• Statistically, we can then peform hypothesis tests
and draw confidence intervals.
Statistical inference
Assume that our statistic of interest is simply average income, and
its estimator follows a normal distribution:
ˆ ~ N(0 , ˆ 2 )
Statistical inference
A centred and normalised distribution can be obtained:
z
ˆ  0
~ N(0,1)
ˆ 
Hypothesis testing
There are three types of hypotheses that can be tested:
1.
An index is equal to a given value:
•
•
2.
An index is higher than a given value:
•
3.
Difference in poverty equals 0
Inequality equals to 20%
Inequality has increased between two periods.
An index is lower than a given value:
•
Poverty has increased between two periods.
The interest of the statistical inferences
•
The outcome of an hypothesis test is a statistical decision
•
The conclusion of the test will either be to reject a null
hypothesis, H0 in favour of an alternative, H1, or to fail to reject it.
•
Most hypothesis tests involving an unknown true population
parameter  fall into three special cases:
1.
2.
3.
H0 : μ = μ0
H0 : μ ≤ μ0
H0 : μ ≥ μ0
against H1: μ ≠ μ0
against H1: μ > μ0
against H1: μ < μ0
The interest of the statistical inferences
•
The ultimate statistical decision may be correct or incorrect. Two
types of error can occur:
Type I error, occurs when we reject H0 when it is in fact true;
•
Type II error, occurs when we fail to reject H0 when H0 is in fact
false.
•
Power of the test of an hypothesis H0 versus H1 is the probability
of rejecting H0 in favour of H1 when H1 is true.
•
P-value is the smallest significance level for which H0 would be
rejected in favour of some H1.
Hypothesis tests
Reject H0: μ = μ0 versus H1: μ ≠ μ0
if and only if :
0  ˆ 0  ˆ  z1 / 2 or 0  ˆ 0  ˆ  z1 / 2
Hypothesis tests
Reject H0: μ ≤ μ0 versus H1: μ > μ0 if and only if :
0  ˆ 0  ˆ  z1 / 2
Hypothesis tests
Reject H0: μ > μ0 versus H1: μ ≤ μ0 if and only if :
0  ˆ 0  ˆ  z  ˆ 0  ˆ  z1
Confidence intervals
•
•
Loosely speaking, a confidence interval contains all of the values
that “cannot be rejected” in a null hypothesis.
Three types of confidence intervals can be drawn: