Homework Assignment

1. Homework Assignment 10.17, 10.19, 10.23, 10.29, 10.32 Due in Class Dec 1

2. Last Time:Finished Contingency TablesReviewed Basics on Linear Regression

4. Suppose I sample n many people: • How many observations do I expect to get in cell (i,j)? • If the Null Hypothesis holds, i.e., if the columns and rows are independent, then I expect the number of observations in cell (i,j) to be How to compare?? Both are unknown!

5. Suppose I sample n many people: How to compare?? Both are unknown!

6. Suppose I sample n many people:

7. Finding a p-value for a Chi-Square Statistic (one-tailed) Table F (T-20)

8. Example: Development in 1st grade

9. Example: Development in 1st grade

10. FIXED

11. Slight Change of Notation Homogeneity of parallel samples

12. Example 255 32 FIXED Equivalent Equations

13. More Generally: The rest is the same as in previous scenario, i.e., we get the same Chi-square again.

16. Special Case: 2x2 Tables

17. The Square of a Standard Normal Random Variableis a Chi-Square Random Variablewith 1 degree of freedom.

18. Today:From descriptive to inference statistics…Estimation and Hypothesis Testingfor Linear Regression

19. Statistical Inference (for a single variable) Estimation: (Confidence Intervals) Point estimate critical value Std. dev. of point estimate ± · For instance: Confidence Interval for the mean: ± ·

20. Statistical Inference (for a single variable) Hypothesis Testing:

21. Example 20 kindergarteners 1pt 2pts 3pts “Popularity Score” = Average Score

22. Example 20 kindergarteners “Social Competence Score”

23. Example 20 kindergarteners “Popularity Score” “Social Competence Score”

24. Statistical Inference (for two variables) Example: Children X: Popularity, Y: Social competence Goal: Explain (linear) relationship between X and Y

25. Statistical Inference (for two variables)

26. Y 1 X

27. Simple (linear) regression Explain the (linear) relationship (if it exists) between random variable X and random variable Y.

28. Four assumptions about the error term 4 For different values of X, the error terms are uncorrelated 1 3 2 The error term is a normally distributed random variable No matter what value X takes, the error has a mean of zero

29. First Assumption:Error has a Normal Distribution Error

30. Second assumption: Average error is zero for each value of X 1 X

31. 3rd assumption: Error has same standard deviation for each value of X Y Error X

32. Y X

33. Y X

34. We will sample data to estimate the parameters. This leads to point estimates, confidence intervals and hypothesis testing for each parameter, in addition to a general test of the model as a whole.

35. Estimation of Intercept and Slope:(just a change of notation)

36. Estimated Regression Line

39. Parameter Estimates: Degrees of freedom loose 1 df for X loose 1 df for Y

40. Recall: Point Estimates (Sample Statistics) are Random Variables

41. Recall: Point Estimates (Sample Statistics) are Random Variables Sampling Distributions

42. Recall: Point Estimates (Sample Statistics) are Random Variables Don’t Know! Hypothesis Testing

43. Remember the general rule for Confidence Intervals: Point estimate critical value Std. dev. of point estimate ± ·

44. Confidence Intervals for Intercept and Slope Point estimate critical value Std. dev. of point estimate ± ·

45. 95% Confidence Intervals for Intercept and Slope

46. 95% Confidence Intervals for Intercept and Slope

47. Hypothesis Test on Slope If p-value of the standardized statistic <  then reject H0 and conclude that there is indeed a linear relationship

48. Hypothesis Test on Slope

49. Computer Output(Note: Different programs differ in style and content!) p-value < .001

50. Analysis of Variancefor Regression How much y differs from mean How much predicted y differs from mean residual / error Involves only data