Least Squares Regression - AP Statistics

1. Which type of plot is used to assess residuals in regression?

   Residual plot. Plots residuals vs. fitted values or explanatory variable.

2. Identify the formula for the regression sum of squares $SS_{reg}$.

   $SS_{reg} = \sum (\hat{y}_i - \bar{y})^2$. Variation in predicted values around mean of $y$.

3. What condition does the normal probability plot of residuals check?

   Normality of residuals. Tests if residuals follow normal distribution.

4. What is the primary assumption about errors in least squares regression?

   Errors are normally distributed. Required for valid inference and prediction intervals.

5. What is the purpose of transforming variables in regression?

   To achieve linearity. Makes curved relationships appear linear.

6. In regression, what is the significance of a horizontal line as the regression line?

   No linear relationship between variables. Slope is zero when variables are uncorrelated.

7. What does a residual plot showing random scatter indicate?

   Appropriate model for the data. Random scatter indicates model assumptions are met.

8. In least squares regression, what does $a$ represent?

   Y-intercept of the regression line. Value of $y$ when $x = 0$.

9. What is the condition of homoscedasticity in regression?

   Constant variance of residuals. Residual spread remains constant across all fitted values.

10. State the condition for using least squares regression.

    Linear relationship between variables. Assumes straight-line relationship exists.

11. What does the term 'extrapolation' refer to in regression?

    Predicting outside the range of the data. Risky because relationships may change outside observed range.

12. What is the formula for the least squares regression line?

    $y = a + bx$. Standard form where $a$ is intercept and $b$ is slope.

13. Find the slope of the regression line given $\sum (x_i - \bar{x})(y_i - \bar{y}) = 10$ and $\sum (x_i - \bar{x})^2 = 5$.

    $b = 2$. Using formula $b = \frac{10}{5} = 2$.

14. What does $R^2$ represent in regression analysis?

    Proportion of variance explained by the model. Coefficient of determination; ranges from 0 to 1.

15. What is the effect of heteroscedasticity in regression?

    Non-constant variance of residuals. Residual spread changes systematically with fitted values.

16. What does the term 'least squares' refer to in regression?

    Minimizing the sum of squared residuals. Method minimizes sum of squared prediction errors.

17. What is the effect of a high leverage point in regression?

    Can disproportionately influence the regression line. Points far from $\bar{x}$ have greater impact.

18. What is the objective of least squares regression?

    Minimize the sum of squared residuals. Finds line that best fits data by minimizing prediction errors.

19. Which term describes the strength and direction of a linear relationship?

    Correlation coefficient. Ranges from -1 to +1, denoted by $r$.

20. Identify the formula for the residual sum of squares $SS_{res}$.

    $SS_{res} = \sum (y_i - \hat{y}_i)^2$. Variation not explained by the regression model.

21. What does it mean if $R^2 = 1$ in regression analysis?

    Perfect explanatory power in the model. Model explains all variation in $y$ perfectly.

22. What does a positive slope indicate in a regression line?

    Positive relationship between variables. As $x$ increases, $y$ tends to increase.

23. Identify the formula for calculating the intercept $a$ in regression.

    $a = \bar{y} - b\bar{x}$. Ensures line passes through point $(\bar{x}, \bar{y})$.

24. What is the effect of an outlier on a regression line?

    Can significantly alter slope and intercept. Outliers pull the line toward themselves disproportionately.

25. Find the intercept $a$ given $\bar{y} = 10$, $b = 2$, and $\bar{x} = 3$.

    $a = 4$. Using formula $a = 10 - (2)(3) = 4$.

26. How do you interpret a residual plot that shows a pattern?

    Model may not be appropriate. Patterns suggest violations of linearity assumption.

27. What does a negative slope indicate in a regression line?

    Negative relationship between variables. As $x$ increases, $y$ tends to decrease.

28. Identify the formula for the total sum of squares $SS_{total}$.

    $SS_{total} = \sum (y_i - \bar{y})^2$. Total variation in $y$ around its mean.

29. What is a residual in regression analysis?

    Difference between observed and predicted value. Measures how far each point is from the regression line.

30. What does it mean if $R^2 = 0$ in regression analysis?

    No explanatory power in the model. Model explains none of the variation in $y$.

31. In least squares regression, what does $b$ represent?

    Slope of the regression line. Rate of change in $y$ per unit change in $x$.

32. Identify the formula for calculating the correlation coefficient $r$.

    $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$. Standardized covariance measuring linear association strength.

33. How do you interpret an $R^2$ value of 0.85?

    85% of variance is explained by the model. Strong model fit with good predictive power.

34. Identify the formula for calculating the slope $b$ in regression.

    $b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$. Covariance divided by variance of $x$.

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