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What does covariance tell you? What it is a function of? What coefficient tells you the strength of the relationship? What is confidence a function of?. Review. What is the central limit theorem? What is a normal distribution? What inference does the central limit theorem help us with?.

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Review

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  1. What does covariance tell you?What it is a function of?What coefficient tells you the strength of the relationship?What is confidence a function of? Review

  2. What is the central limit theorem?What is a normal distribution?What inference does the central limit theorem help us with? Review of central limit theorem

  3. yi = a + bxi + eiyhat = a + bx Two formulas • What is yhat? • What is yi? • What is xi? • What is ei? • What is b? • What is a?

  4. Interpreting results What is the difference between b and Beta? What is the standard error How do you compute t? What is the significance level?

  5. Residual review What is a residual? What is the mean of residuals? What assumption do we make about residuals?

  6. What is a z score?How is it computed?What is the beta coefficient?How is it different from the b in terms of interpreting the effect? Z score review

  7. T statistic review • What is the formula for the t statistic? • If the t = 2, how confident are we? • (what are we confident about?)

  8. The intercept? • If the intercept is 3, and the dependent variable ranges from 1-4 and the independent variable is 1-4, what other information do we need to know the value of the DV when the IV is at its lowest value? • The slope is 2. • What is the value of the DV when the IV is at its lowest value?

  9. If you multiply the dependent variable by 100, what numbers change? How do they change? * • What numbers do not change? * *Potential answers: B, Beta, standard error, t, significance

  10. Where does our estimate of the error come from? • The residuals. If the points are far from the slope, then we are less confident. • If the points are close to the slope, then we are more confident.

  11. Can we be wrong about rejecting a null hypothesis?There are two kinds of errors: • (Type 1) a true null hypothesis can be incorrectly rejected • (Type 2) a false null hypothesis can fail to be rejected.

  12. Type 2 error is more serious • We you fail to reject the hypothesis, you do not prove the hypothesis is wrong. (remember, we don’t ever prove anything). • It could be measurement error and all kinds of statistical problems that lead to rejecting a null hypothesis.

  13. Null Hypothesis Rejected • If you reject it, then you have tried to prove your theory wrong and you could not. • Don’t forget that you haven’t proven anything (we never prove anything) • You still have other ways of trying to prove it wrong

  14. What is the question that we ask in statistical analysis? • How much better have we done than the mean in predicting values of y from x?

  15. How do we know we have done better than the mean? • Distance between the slope and the mean is great • What is the confidence of “doing better than the mean” likely determined by? • Ratio of explained to unexplained variance

  16. Wouldn’t it be great to have a coefficient that told us the ratio of explained to unexplained variance? • Total Variance • = • Explained Variance • + • Unexplained variance

  17. R square • R square = Explained Variance Unexplained + Explained Variance Unexplained Variance + Explained Variance = what? (total variance)

  18. For each observation, you calculate the distance from the mean to the slope squared to get explained variance. • Then divide by the total sum of squares, which is total variance

  19. Pearson r and R square • Pearson r squared is the same as R square • (in the bivariate case – one independent variable) (Pearson r)2 = R square R square is standardized and symmetric Symmetric means that it doesn’t matter which is the independent variable and which is the dependent variable

  20. An Example

  21. regr happy prestg80, beta Source | SS df MS Number of obs = 11 -------------+------------------------------ F( 1, 9) = 3.30 Model | 1.31753739 1 1.31753739 Prob > F = 0.1026 Residual | 3.59155351 9 .399061502 R-squared = 0.2684 -------------+------------------------------ Adj R-squared = 0.1871 Total | 4.90909091 10 .490909091 Root MSE = .63171 ------------------------------------------------------------------------------ happy | Coef. Std. Err. t P>|t| Beta -------------+---------------------------------------------------------------- prestg80 | -.0380391 .0209348 -1.82 0.103 -.518061 _cons | 3.330371 .8050567 4.14 0.003 . ------------------------------------------------------------------------------

  22. Formula for the slope(in the bivariate case)

  23. Formula for r and beta* * Beta is the same as r in the case of bivariate

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