1. PEP-PMMA Training Session Statistical inference Lima, Peru Abdelkrim Araar / Jean-Yves Duclos 9-10 June 2007

2. 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.

3. Statistical inference Assume that our statistic of interest is simply average income, and its estimator follows a normal distribution:

4. Statistical inference A centred and normalised distribution can be obtained:

5. Hypothesis testing There are three types of hypotheses that can be tested: 1.An index is equal to a given value: Difference in poverty equals 0 Inequality equals to 20% 2.An index is higher than a given value: Inequality has increased between two periods. 3.An index is lower than a given value: Poverty has increased between two periods.

6. 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, H 0 in favour of an alternative, H 1, or to fail to reject it. Most hypothesis tests involving an unknown true population parameter  fall into three special cases: 1.H 0 : μ = μ 0 against H 1 : μ ≠ μ 0 2.H 0 : μ ≤ μ 0 against H 1 : μ > μ 0 3.H 0 : μ ≥ μ 0 against H 1 : μ < μ 0

7. 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 H 0 when it is in fact true; Type II error, occurs when we fail to reject H 0 when H 0 is in fact false. Power of the test of an hypothesis H 0 versus H 1 is the probability of rejecting H 0 in favour of H 1 when H 1 is true. P-value is the smallest significance level for which H 0 would be rejected in favour of some H 1.

8. Hypothesis tests Reject H 0 : μ = μ 0 versus H 1 : μ ≠ μ 0 if and only if :

9. Hypothesis tests Reject H 0 : μ ≤ μ 0 versus H 1 : μ > μ0 if and only if :

10. Hypothesis tests Reject H 0 : μ > μ 0 versus H 1 : μ ≤ μ0 if and only if :

11. 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: