Class 2 Statistical Inference

1. SKEMA Ph.D programme 2010-2011 Class 2Statistical Inference Lionel Nesta Observatoire Français des Conjonctures Economiques Lionel.nesta@ofce.sciences-po.fr

2. Hypothesis Testing

3. The Notion of Hypothesis in Statistics • Expectation • An hypothesis is a conjecture, an expected explanation of why a given phenomenon is occurring • Operational -ity • An hypothesis must be precise, univocal and quantifiable • Refutability • The result of a given experiment must give rise to either the refutation or the corroboration of the tested hypothesis • Replicability • Exclude ad hoc, local arrangements from experiment, and seek universality

4. Examples of Good and Bad Hypotheses  « The stakes Peugeot and Citroen have the same variance » «  God exists! » « In general, the closure of a given production site in Europe is positively associated with the share price of a given company on financial markets. » « Knowledge has a positive impact on economic growth »   

5. Hypothesis Testing • In statistics, hypothesis testing aims at accepting or rejecting a hypothesis • The statistical hypothesis is called the “null hypothesis” H0 • The null hypothesis proposes something initially presumed true. • It is rejected only when it becomes evidently false, that is, when the researcher has a certain degree of confidence, usually 95% to 99%, that the data do not support the null hypothesis. • The alternative hypothesis (or research hypothesis) H1 is the complement of H0.

6. Hypothesis Testing • There are two kinds of hypothesis testing: • Homogeneity test compares the means of two samples. • H0 : Mean(x) = Mean(y) ; Mean(x) = 0 • H1 : Mean(x) ≠ Mean(y) ; Mean(x) ≠ 0 • Conformity test looks at whether the distribution of a given sample follows the properties of a distribution law (normal, Gaussian, Poisson, binomial). • H0 : ℓ(x) = ℓ*(x) • H1 : ℓ(x) ≠ ℓ*(x)

7. The Four Steps of Hypothesis Testing • Spelling out the null hypothesis H0 et and the alternative hypothesis H1. • Computation of a statistics corresponding to the distance between two sample means (homogeneity test) or between the sample and the distribution law (conformity test). • Computation of the (critical) probability to observe what one observes. • Conclusion of the test according to an agreed threshold around which one arbitrates between H0 and H1 .

8. The Logic of Hypothesis Testing • We need to say something about the reliability (or representativeness) of a sample mean • Large number theory; Central limit theorem • The notion of confidence interval • Once done, we can whether two mean are alike • If so (not), their confidence intervals are (not) overlapping

9. Statistical Inference • In real life calculating parameters of populations is prohibitive because populations are very large. • Rather than investigating the whole population, we take a sample, calculate a statistic related to the parameter of interest, and make an inference. • The sampling distribution of the statistic is the tool that tells us how close is the statistic to the parameter.

10. Prerequisite 1 Standard Normal Distribution

11. The Standard Normal Distribution The standard normal distribution, also called Z distribution, represents a probability density function with mean μ = 0 and standard deviationσ = 1. It is written as N(0,1).

12. The Standard Normal Distribution Since the standard deviation is by definition 1, each unit on the horizontal axis represents one standard deviation

13. The Standard Normal Distribution Because of the shape of the Z distribution (symmetrical), statisticians have computed the probability of occurrence of events for given values of z.

14. The Standard Normal Distribution 68% of observations 95% of observations 99.7% of observations

15. The Standard Normal Distribution 95% of observations 2.5% 2.5%

16. The Standard Normal Distribution (z scores) P(Z ≥ 0) P(Z < 0)

17. Probability of an event (z = 0.51) P(Z ≥ 0.51)

18. = ?? Probability of an event (z = 0.51) • The z-score is used to compute the probability of obtaining an observed score. • Example • Let z = 0.51. What is the probability of observing z=0.51? • It is the probability of observing z ≥ 0.51: P(z ≥ 0.51)

19. Standard Normal Distribution Table

20. Probability of an event (Z = 0.51) • The Z-score is used to compute the probability of obtaining an observed score. • Example • Let z = 0.51. What is the probability of observing z=0.51? • It is the probability of observing z ≥ 0.51: P(z ≥ 0.51) • P(z ≥ 0.51) = 0.3050

21. Prerequisite 2 Normal Distribution

22. Normal Distributions Normal distributions are just like standard normal distributions (or z distributions) with different values for the mean μ and standard deviationσ. This law is written N (μ,σ²). The normal distribution is symmetrical.

23. The Normal Distribution In probability, a random variable follows a normal distribution law (also called Gaussian, Laplace-Gauss distribution law) of mean μ and standard deviation σ if its probability density function is such that This law is written N (μ,σ ²). The density function of a normal distribution is symmetrical.

24. Normal distributions for different values of μ and σ

25. Standardization of Normal Distributions Still, it would be nice to be able to say something about these distributions just like we did with the z distribution. For example, textile companies (and clothes manufacturers) may be very interested in the distribution of heights of men and women, for a given country (provided that we have all observations). How could we compute the proportion of men taller than 1.80 meters?

26. Standardization of Normal Distributions Assuming that the heights of men is distributed normal, is there any way we could express it in terms of a z distribution? We must center the distribution around 0. We must express any value in terms of deviation around the mean : (X – μ) We must express (or reduce) each deviation in terms of number of standard deviation σ. (X – μ) / σ

27. Standardization of Normal Distributions Standardization of a normal distribution is the operation of recovering a z distribution from any other distribution, assuming the distribution is normal. It is achieved by centering (around the mean) and reducing (in terms of number of standard deviations) each observation. The obtained z value expresses each observation by its distance from the mean, in terms of number of standard deviations.

28. Example • Suppose that for a population of students of a famous business school in Sophia-Antipolis, grades are distributed normal with an average of 10 and a standard deviation of 3. What proportion of them • Exceeds 12 ; Exceeds 15 • Does not exceed 8 ; Does not exceed 12 • Let the mean μ = 10 and standard deviation σ = 3:

29. Implication 1Intervals of likely values

30. Inverting the way of thinking • Until now, we have thought in terms of observations x and mean μ and standard deviation σ to produce the z score. • Let us now imagine that we do not know x, we know μ and σ. If we consider any interval, we can write:

31. Inverting the way of thinking • If z∈[-2.55;+2.55] we know that 99% of z-scores will fall within the range • If z∈[-1.64;+1.64] we know that 90% of z-scores will fall within the range • Let us now consider an interval which comprises 95% of observations. Looking at the z table, we know that z=1.96

32. Example • Take the population of students of this famous business school in Sophia-Antipolis, with average of 10 and a standard deviation of 3. What is the 99% interval ? 95% interval? 90% interval?

33. Prerequisite 3 Sampling theory

34. Why worrying about sampling theory? The social scientist is not so much interested in the characteristics of the sample itself. Most of the time, the social scientist wants to say something about the population itself looking at the sample. In other words, s/he wants to infer something about the population from the sample.

35. On the use of random samples The quality of the sample is key to statistical inference. The most important thing is that the sample must be representative of the characteristics of the population. The means by which representativeness can be achieved is by drawing random samples, where each individual observation have equal probability to be drawn. Because we would be inferring wrong conclusions from biased samples, the latter are worse than no sample at all.

36. Use of random samples The quality of the sample is key to statistical inference. The most important thing is that the sample must be representative of the characteristics of the population. The means by which representativeness can be achieved is by drawing random samples, where each individual observation have equal probability to be drawn. Hence observations are mutually independent. Because we would be inferring wrong conclusions from biased samples, the latter are worse than no sample.

37. Reliability of random samples The ultimate objective with the use of random samples is to infer something about the underlying population. Ideally, we want the sample mean to be as close as possible to the population mean μ. In other words, we are interested in the reliability of the sample. The are two ways to deal with reliability: Monte Carlo Simulation (infinite number of samples) Sampling theory (moments of a distribution)

38. Moment 1 – The Mean Our goal is to estimate the population mean μ from the sample mean . How is the sample mean a good estimator of the population mean ? Reminder : the sample mean is computed as follows. The trick is to consider each observations as a random variable, in line with the idea of a random sample.

39. Moment 1 – The Mean What is the expected value of Xi – E(Xi) – if I draw it an infinite number of times ? Obviously if samples are random, then the expected value of Xi is μ. … working out the math… On average, the sample mean will be on target, that is, equal to the population mean.

40. Moment 2 – The Variance Doing just the same with the variance. We simply need to know that if two variables are independent, then the following holds: The standard deviation of the sample means represents the estimation error of approximation of the population mean by the sample mean, and therefore it is called the standard error.

41. Forms of sampling distributions • With random samples, sample means X vary around the population mean μ with a standard deviation of σ/√n (the standard error). • Large number theory tells us that the sample mean will converge to the population (true) mean as the sample size increases. • But what about the shape of the distribution, essential if we want to use z-scores?! The shape of the distribution will be normally distributed, regardless of the form of the underlying distribution of the population, provided that the sample size is large enough. • Central Limit Theorem tells us that for many samples of like and sufficiently large size, the histogram of these sample means will appear to be a normal distribution.

42. The Dice Experiment

43. The Dice Experiment (n = 2)

44. 6/36 5/36 4/36 3/36 2/36 1/36 1 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

45. From SKEMA sample grade distribution…

46. …to SKEMA sample mean distribution

47. …to SKEMA sample mean distribution From SKEMA sample grade distribution… …to SKEMA sample mean distribution

48. …to SKEMA sample mean distribution Note the change in horizontal axis !!

49. Implication 2Confidence Interval

50. Confidence Interval • In statistics, a confidence interval is an interval within which the value of a parameter is likely to be (the unknown population mean). Instead of estimating the parameter by a single value, an interval of likely estimates is given. • Confidence intervals are used to indicate the reliability of an estimate. • Reminder 1. The sample mean is a random variable following a normal distribution • Reminder 1. The sample values X and σs can be used to approximate the population mean μ and its s.d. on σp.