1. Statistical inference

2. Statistical inference
Definition : generalization from a sample to a population.
2 cases:
Is a sample belongs to an hypothetical population?
Is two samples belong to the same hypothetical population?

3. Statistical inference
1st possibility x ?
Inference

4. Statistical inference
2nd possibility x ?
Inference

5. Hypotheses We test H0 H0= Null hypothesis H1 = Alternative hypothesis
m = Mean of the population
k = Constant 1
H0= Null hypothesis
H1 = Alternative hypothesis
m1 = Mean of the first population
m2 = Mean of the second population 2
We test H0

6. Decision
From the sample(s) we decide if we reject or not the null hypothesis.
When we are doing inference we are never certain that we took the right decision
Population
Sample
Decision
Identical
Different
Good
Error 2
Error 1

7. Decision 2 type of errors:
1 – If we inferred that 2 groups belong to two different populations when they don’t. We rejected H0 when H0 was true.
2 – If we inferred that 2 groups belong to the same population when they don’t. We kept H0 when H0 was false.
Population
Sample
Decision
Identical
Different
Good
Error 2
Error 1

8. Sampling Distribution of the Mean
1- Inference about the mean of a population Sampling Distribution of the Mean
Sample (n)
Sampling Distribution of the Mean
Population

9. Sampling Distribution of the Mean
Characteristics:
Follows a normal curve.
The mean will be equal to the one of the population
The standard deviation will be equal to
The larger the sample size is, the smaller the standard error will be.

10. Sampling Distribution of the Mean
Sample
Sampling Distribution of the Mean
Population

11. Sampling Distribution of the Mean
Sample
Sampling Distribution of the Mean
Population

12. Sampling Distribution of the Mean
Sample
Sampling Distribution of the Mean
Population

13. Sampling Distribution of the Mean
Sample
Sampling Distribution of the Mean
Population

14. Test of Significance
If we suppose that the null hypothesis is true, what is the probability of observing the giving sample mean?
If it is unlikely, we will reject H0, else we will keep H0.
Unlikely: 5% or 1% = a = significance threshold

15. Test of Significance Example: one side H0: m = 72
H1: m < 72 (based on previous studies)
a = 0.05 (5%)
s = 9
= 65
n = 36
Because zx is greater za we reject the null hypothesis and accept the alternative hypothesis
za = 1.65

16. Test of Significance Example : 2 sides H0: m = 72 H1: m  72
= 68
n = 36
Because zx is greater za we reject the null hypothesis and accept the alternative hypothesis
za = 1.96

17. Confidence intervals
We are never sure that the mean of our sample is exactly the real mean of the population. Therefore, instead of given the mean only, it is possible de quantify our level of certitude by specifying a confidence interval around the mean.

18. Confidence intervals Example: CI = 95% = 50,7 n = 100 s = 20
Therefore, there is a 95% probability that the mean of the population is between and 54.62
za = 1.96

19. Confidence intervals Example: CI = 99% = 50,7 n = 100 s = 20
Therefore, there is a 99% probability that the mean of the population is between and 55.86
za = 2.58

20. Distribution of sample mean differences
2- Inference for the difference  between two population means distribution of sample mean differences
Samples (n)
Distribution of sample mean differences
Population

21. Distribution of sample mean differences
Characteristics:
Follows a normal distribution
The mean will be equal to 0 (m1-m2=0)
The standard deviation will be equal to:

22. Decision rule

23. Test of Significance
Example: What is the probability of observed difference between the following groups?
H0: m1 = m2 (m1 - m2 = 0)
H1: m1  m2 (m1 - m2  0)
a = 0.05 (5%)
= 50
s1 = 5
n1 = 36
= 48
s2 = 5
n2 = 36

24. Test of Significance
Example: What is the probability of observed difference between the following groups?
H0: m1 = m2 (m1 - m2 = 0)
H1: m1  m2 (m1 - m2  0)
a = 0.05 (5%)
= 50
s1 = 5
n1 = 36
= 48
s2 = 5
n2 = 36
Because the observed z is lower than the critical (za) we will keep the null hypothesis

25. Confidence intervals

26. Test of Significance Example: a 95% confidence interval
H0: m1 = m2 (m1 - m2 = 0)
H1: m1  m2 (m1 - m2  0)
a = 0.05 (5%)
= 50
s1 = 5
n1 = 36
= 48
s2 = 5
n2 = 36

27. Test of Significance Example: a 95% confidence interval
H0: m1 = m2 (m1 - m2 = 0)
H1: m1  m2 (m1 - m2  0)
a = 0.05 (5%)
= 50
s1 = 5
n1 = 36
= 48
s2 = 5
n2 = 36
Therefore there is a 95% probability that the mean difference between the populations is between
and