1. Inferential Statistics 2 Maarten Buis January 11, 2006

2. outline Student Recap Sampling distribution Hypotheses Type I and II errors and power testing means testing correlations

3. Sampling distribution PrdV example from last lecture. If H 0 is true, than the population consists of 16 million persons of which 41% (=6.56 million persons) supports de PrdV. I have drawn 100,000 random samples of 2,598 persons each and compute the average support in each sample.

4. Sampling distribution 5% or 50,000 samples have a mean of 39% or less. So if we reject H 0 when we find a support of 39% or less than we will have a 5% chance of making an error. Notice: We assume that the only reason we would make an error is random sampling error.

6. More precise approach We want to know the score below which only 5% of the samples lie. Drawing lots of random samples is a rather rough approach, an alternative approach is to use the theoretical sampling distribution. The proportion is a mean and the sampling distribution of a mean is the normal distribution with a mean equal to the H 0 and a standard deviation (called standard error) of

7. More precise approach For a standard normal distribution we know the z-score below which 5% of the samples lie (Appendix 2, table A): -1.68 So if we compute a z-score for the observed value (.31) and it is below -1.68 we can reject the H 0, and we will do so wrongly in only 5% of the cases

8. More precise approach  is the mean of the sampling distribution, so.41 (H 0 ) se is,  of a proportion is so the se is so the z-score is -10.4 is less than -1.68, so we reject the H 0

9. Null Hypothesis A sampling distribution requires you to imagine what the population would look like if H 0 is true. This is possible if H 0 is one value (41%) This is impossible if H 0 is a range (<41%) So H 0 should always contain a equal sign (either = or ≤ or ≥)

10. Null hypothesis In practice the H 0 is almost always 0, e.g.: –difference between two means is 0 –correlation between two variables is 0 –regression coefficient is 0 This is so common that SPSS always assumes that this is the H 0.

11. Undirected Alternative Hypotheses Often we have an undirected alternative hypothesis, e.g.: –the difference between two means is not zero (could be either positive or negative) –the correlation between two variables is not zero (could be either positive or negative) –the regression coefficient is not zero (could be either

12. Directed alternative hypothesis In the PrdV example we had a directed alternative hypothesis: Support for PrdV is less than 41%, since PrdV would have still participated if his support were more than 41%.

13. Type I and Type II errors actual situation decisionH 0 is TrueH 0 is False reject H 0 Type I error probability =  correct decision probability = 1-  (power) do not reject H 0 correct decision probability = 1-  Type II error probability = 

14. Type I error rate You choose the type I error rate (  ) It is independent of sample size, type of alternative hypothesis, or model assumptions.

15. Type I versus type II error rate a low probability of rejecting H 0 when H 0 is true (type I error), is obtained by: rejecting the H 0 less often, Which also means a higher probability of not rejecting H 0 when H 0 is false (type II error), In other words: a lower probability of finding a significant result when you should (power).

16. How to increase your power: Lower type I error rate Larger sample size Use directed instead of undirected alternative hypothesis Use more assumptions in your model (non- parametric tests make less assumptions, but are also have less power)

17. Testing means What kind of hypotheses might we want to test: –Average rent of a room in Amsterdam is 300 euros –Average income of males is equal to the average income of females

18. Z versus t In the PrdV example we knew everything about the sampling distribution with only an hypothesis about the mean. In the rent example we don’t: we have to estimate the standard deviation. This adds uncertainty, which is why we use the t distribution instead of the normal Uncertainty declines when sample size becomes larger. In large samples (N>30) we can use the normal.

19. t-distribution It has a mean and standard error like the normal distribution. It also has a degrees of freedom, which depends on the sample size The larger the degrees of freedom the closer the t-distribution is to the normal distribution.

20. Data: rents of rooms rent room 1175room 11240 room 2180room 12250 room 3185room 13250 room 4190room 14280 room 5200room 15300 room 6210room 16300 room 7210room 17310 room 8210room 18325 room 9230room 19620 room 10240

21. Rent example H 0 :  =300, H A :  ≠ 300 We choose  to be 5% N = 19, so df= 18 We reject H 0 if we find a t less than -2.101 or more than 2.101 (appendix B, table 2) We do not reject H 0 if we find a t between -2.101 and 2.101.

22. Rent example We use s 2 as an estimate of  2 So -1.85 is between -2.101 and 2.101, so we do not reject H 0

23. Compare means in SPSS

25. Do before Monday Read Chapter 9 and 10 Do the “For solving Problems”