Linear Regression Models - AP Statistics
Card 1 of 30
Identify the limitation of using $R^2$ alone to judge model fit.
Identify the limitation of using $R^2$ alone to judge model fit.
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Does not show causation. High $R^2$ doesn't prove the variables are causally related.
Does not show causation. High $R^2$ doesn't prove the variables are causally related.
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Identify the slope in the regression equation $y = 3 + 2x$.
Identify the slope in the regression equation $y = 3 + 2x$.
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- The slope is the coefficient of $x$ in the linear equation.
- The slope is the coefficient of $x$ in the linear equation.
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What does $R^2$ represent in a linear regression model?
What does $R^2$ represent in a linear regression model?
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Coefficient of determination. Measures proportion of variance in $y$ explained by the model.
Coefficient of determination. Measures proportion of variance in $y$ explained by the model.
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Find the $R^2$ if the correlation $r$ is 0.8.
Find the $R^2$ if the correlation $r$ is 0.8.
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0.64. $R^2 = r^2$, so $(0.8)^2 = 0.64$.
0.64. $R^2 = r^2$, so $(0.8)^2 = 0.64$.
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What does an $R^2$ value of 1 imply about the model?
What does an $R^2$ value of 1 imply about the model?
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Perfect fit. All data points lie exactly on the regression line.
Perfect fit. All data points lie exactly on the regression line.
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Choose the term for residuals in regression analysis.
Choose the term for residuals in regression analysis.
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Errors. Residuals are the differences between observed and predicted values.
Errors. Residuals are the differences between observed and predicted values.
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What does it mean if residuals show a pattern?
What does it mean if residuals show a pattern?
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Model is not appropriate. Patterns suggest non-linear relationship or other violations.
Model is not appropriate. Patterns suggest non-linear relationship or other violations.
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Find the correlation coefficient if $R^2 = 0.81$.
Find the correlation coefficient if $R^2 = 0.81$.
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$\text{either } 0.9 \text{ or } -0.9$. $r = \pm\sqrt{R^2} = \pm\sqrt{0.81} = \pm 0.9$.
$\text{either } 0.9 \text{ or } -0.9$. $r = \pm\sqrt{R^2} = \pm\sqrt{0.81} = \pm 0.9$.
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Identify the dependent variable in regression analysis.
Identify the dependent variable in regression analysis.
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Response variable. The variable being predicted or explained ($y$).
Response variable. The variable being predicted or explained ($y$).
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State the independent variable in the equation $y = 5x + 3$.
State the independent variable in the equation $y = 5x + 3$.
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$x$. The predictor or explanatory variable in the equation.
$x$. The predictor or explanatory variable in the equation.
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What is the significance of the $p$-value in regression?
What is the significance of the $p$-value in regression?
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Tests hypothesis about slope. Tests if slope is significantly different from zero.
Tests hypothesis about slope. Tests if slope is significantly different from zero.
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What does a $p$-value less than 0.05 indicate?
What does a $p$-value less than 0.05 indicate?
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Statistically significant. Evidence to reject null hypothesis at 5% significance level.
Statistically significant. Evidence to reject null hypothesis at 5% significance level.
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Find and correct: $y = 2x + 3$ is a nonlinear model.
Find and correct: $y = 2x + 3$ is a nonlinear model.
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Linear model. $y = 2x + 3$ is linear since it has constant slope.
Linear model. $y = 2x + 3$ is linear since it has constant slope.
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Identify the point where the regression line crosses the y-axis.
Identify the point where the regression line crosses the y-axis.
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Y-intercept. The value of $y$ when $x = 0$ (the constant term).
Y-intercept. The value of $y$ when $x = 0$ (the constant term).
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State the formula to calculate residuals.
State the formula to calculate residuals.
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$e_i = y_i - \bar{y}_i$. Difference between observed and predicted values.
$e_i = y_i - \bar{y}_i$. Difference between observed and predicted values.
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What is the primary purpose of a regression model?
What is the primary purpose of a regression model?
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Predict values of $y$. To estimate or forecast response variable values.
Predict values of $y$. To estimate or forecast response variable values.
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What does extrapolation mean in regression?
What does extrapolation mean in regression?
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Predicting outside data range. Making predictions beyond the range of observed data.
Predicting outside data range. Making predictions beyond the range of observed data.
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What is the impact of high leverage points in regression?
What is the impact of high leverage points in regression?
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Influence regression line. Points far from mean $x$ have greater impact on line.
Influence regression line. Points far from mean $x$ have greater impact on line.
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Calculate the residual for $y_i = 8$ and $\bar{y}_i = 5$.
Calculate the residual for $y_i = 8$ and $\bar{y}_i = 5$.
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- Residual = observed - predicted = $8 - 5 = 3$.
- Residual = observed - predicted = $8 - 5 = 3$.
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What does a correlation coefficient of 0 imply?
What does a correlation coefficient of 0 imply?
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No linear relationship. Zero correlation means no linear association between variables.
No linear relationship. Zero correlation means no linear association between variables.
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What is the formula for calculating the correlation coefficient $r$?
What is the formula for calculating the correlation coefficient $r$?
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$r = \frac{\text{cov}(X, Y)}{\text{std}(X) \times \text{std}(Y)}$. Covariance divided by product of standard deviations.
$r = \frac{\text{cov}(X, Y)}{\text{std}(X) \times \text{std}(Y)}$. Covariance divided by product of standard deviations.
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Determine the effect of adding a constant to all $y$ values.
Determine the effect of adding a constant to all $y$ values.
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Shifts intercept. Adding constant to all $y$ values shifts y-intercept up.
Shifts intercept. Adding constant to all $y$ values shifts y-intercept up.
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What does the slope coefficient represent in a regression model?
What does the slope coefficient represent in a regression model?
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Change in $y$ per unit $x$. How much $y$ changes for each one-unit increase in $x$.
Change in $y$ per unit $x$. How much $y$ changes for each one-unit increase in $x$.
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What does a residual of 0 indicate?
What does a residual of 0 indicate?
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Perfect prediction. Model predicted the exact observed value.
Perfect prediction. Model predicted the exact observed value.
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Find the regression line if $b_0=2$ and $b_1=4$.
Find the regression line if $b_0=2$ and $b_1=4$.
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$y = 2 + 4x$. Substituting values into the standard form equation.
$y = 2 + 4x$. Substituting values into the standard form equation.
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What is the primary purpose of a regression model?
What is the primary purpose of a regression model?
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Predict values of $y$. To estimate or forecast response variable values.
Predict values of $y$. To estimate or forecast response variable values.
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State the condition when a linear model is appropriate.
State the condition when a linear model is appropriate.
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Linear relationship. Data points should follow a straight-line pattern.
Linear relationship. Data points should follow a straight-line pattern.
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Choose the term for residuals in regression analysis.
Choose the term for residuals in regression analysis.
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Errors. Residuals are the differences between observed and predicted values.
Errors. Residuals are the differences between observed and predicted values.
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What is the significance of the $p$-value in regression?
What is the significance of the $p$-value in regression?
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Tests hypothesis about slope. Tests if slope is significantly different from zero.
Tests hypothesis about slope. Tests if slope is significantly different from zero.
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Identify the term for unexplained variability in regression.
Identify the term for unexplained variability in regression.
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Error variance. Variability not explained by the regression model.
Error variance. Variability not explained by the regression model.
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