1. Module 2 Association and Correlation Relationship between 2 quantitative variables: Scatterplot and Correlation Relationship between 2 qualitative variables: Crosstab and Chi-square test

2. Relation between 2 quantitative varibles We want to see if there is any relationship between the results on exams and the number of hours used for studies. Person ABCDEFGHIJ Hours/ Day 4567535788 Result 20252235151422303739

3. Scatter plots are used to describe the relationship between two quantitative variables.

4. Correlation X Y X and Y vary together without any theories that one affects the other Example: Y = price of houses X = prices of apartments Often: Y = dependent variable X = independent variable Example: Y = prices of houses X= interest

5. Coefficient of correlation The coefficient of correlation r is a measure of linear relationship between two variables x and y. The coefficient of correlation can take values between –1 and +1. Be aware that r is a measure of linear relationships. Even if r = 0 there can be a nonlinear relationship between x and y.

6. We want to test: Null hypothesis: There is no linear relationship between the variables x and y. Alternative hypothesis: There is a linear relationship between the variables x and y.

7. Scatterplot: babies from VK Scatterplot: babies from VK

8. Correlation between weight and height of newborn babies Correlation between weight and height of newborn babies Correlations 1,765**,,000 35,765**1,000, 35 Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N LENGTH WEIGHT LENGTHWEIGHT Correlation is significant at the 0.01 level (2-tailed). **.

9. Is there a relationship between mean income level and population in the 50 American states ? Draw a scatter plot! What is the coefficient of correlation?

10. Income vs Population

11. Relationship between 2 qualitative variables Relationship between 2 qualitative variables A study was done to investigate whether the usage of bicycle helmets is an effective way to protect people in bicycle accidents from skull damage. 793 persons participated in the study, with the following results: Observed Crosstable Used a helmet Did not use a helmet Skull damage17218 No skull damage 130428

12. We want to test: Null hypothesis: There is no relationship between skull damages and helmets. (The amount of skull damages is the same no matter a person in an accident is using a helmet or not. ) Alternative hypothesis: There is a relationship between skull damages and helmets. (The amount of skull damages is different for those who use helmets and those who don’t. )

13. We compute the expected value if the null hypothesis is true and perform a Chi-square test : Expected Cross Table Used a helmet Did not use a helmet Skull damage235·147/793= =43,6 235·646/793= 191,4 No skull damage 558·147/793= =103,4 558·646/793= =454,6

14. We compare the observed table with the expected one. If the tables differ much we will reject the null hypothesis. Then we have empirical evidence that there can be a dependency between the variables

15. If the null hypothesis is true we would get a value close to zero. Is 28,57 so far away from zero that we can reject the null hypothesis? We compare the obtained p-value with our chosen level of significance. Observed p-value: 0,000 Conclusion?

16. Since the p-value = 0,000 < 0.05, we can reject the null hypothesis. There is a significant difference between the percentage of people with skull damage in the two groups of bikers.