AP Statistics Flashcards: Linear Regression Models

What this deck covers

This deck focuses on Linear Regression Models, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the regression line if $b_0=2$ and $b_1=4$.

Answer: $y = 2 + 4x$. Substituting values into the standard form equation.

Flashcard 2: What does a positive slope indicate about the relationship between variables?

Answer: Positive correlation. Positive slope means $y$ increases as $x$ increases.

Flashcard 3: What is the interpretation of a slope of 0 in a regression model?

Answer: No linear relationship. Zero slope means $y$ doesn't change as $x$ changes.

Flashcard 4: What is the significance of the $p$-value in regression?

Answer: Tests hypothesis about slope. Tests if slope is significantly different from zero.

Flashcard 5: Calculate the residual for $y_i = 8$ and $\bar{y}_i = 5$.

Answer:

1. Residual = observed - predicted = $8 - 5 = 3$.

Flashcard 6: What is the formula for calculating the correlation coefficient $r$?

Answer: $r = \frac{\text{cov}(X, Y)}{\text{std}(X) \times \text{std}(Y)}$. Covariance divided by product of standard deviations.

Flashcard 7: Identify the y-intercept in $y = -2x + 6$.

Answer:

1. Y-intercept is the constant term when $x=0$.

Flashcard 8: What is the primary purpose of a regression model?

Answer: Predict values of $y$. To estimate or forecast response variable values.

Flashcard 9: What does an $R^2$ value of 1 imply about the model?

Answer: Perfect fit. All data points lie exactly on the regression line.

Flashcard 10: State the condition when a linear model is appropriate.

Answer: Linear relationship. Data points should follow a straight-line pattern.

Flashcard 11: Identify the limitation of using $R^2$ alone to judge model fit.

Answer: Does not show causation. High $R^2$ doesn't prove the variables are causally related.

Flashcard 12: What is the formula for calculating the correlation coefficient $r$?

Answer: $r = \frac{\text{cov}(X, Y)}{\text{std}(X) \times \text{std}(Y)}$. Covariance divided by product of standard deviations.

Flashcard 13: What is the interpretation of a slope of 0 in a regression model?

Answer: No linear relationship. Zero slope means $y$ doesn't change as $x$ changes.

Flashcard 14: Find the $R^2$ if the correlation $r$ is 0.8.

Answer: 0.64. $R^2 = r^2$, so $(0.8)^2 = 0.64$.

Flashcard 15: What is the primary purpose of a regression model?

Answer: Predict values of $y$. To estimate or forecast response variable values.

Flashcard 16: What is the formula for the least squares regression line?

Answer: $y = b_0 + b_1x$. Standard form where $b_0$ is y-intercept and $b_1$ is slope.

Flashcard 17: Calculate the residual for $y_i = 8$ and $\bar{y}_i = 5$.

Answer:

1. Residual = observed - predicted = $8 - 5 = 3$.

Flashcard 18: What does an $R^2$ value of 1 imply about the model?

Answer: Perfect fit. All data points lie exactly on the regression line.

Flashcard 19: What does the slope coefficient represent in a regression model?

Answer: Change in $y$ per unit $x$. How much $y$ changes for each one-unit increase in $x$.

Flashcard 20: Choose the term for residuals in regression analysis.

Answer: Errors. Residuals are the differences between observed and predicted values.

Flashcard 21: Determine the effect of adding a constant to all $y$ values.

Answer: Shifts intercept. Adding constant to all $y$ values shifts y-intercept up.

Flashcard 22: Find the predicted value of $y$ for $x=5$ in $y = 3x + 4$.

Answer:

1. Substitute $x=5$: $y = 3(5) + 4 = 19$.

Flashcard 23: What does the slope coefficient represent in a regression model?

Answer: Change in $y$ per unit $x$. How much $y$ changes for each one-unit increase in $x$.

Flashcard 24: State the formula to calculate residuals.

Answer: $e_i = y_i - \bar{y}_i$. Difference between observed and predicted values.

Flashcard 25: State the independent variable in the equation $y = 5x + 3$.

Answer: $x$. The predictor or explanatory variable in the equation.

Flashcard 26: What does a $p$-value less than 0.05 indicate?

Answer: Statistically significant. Evidence to reject null hypothesis at 5% significance level.

Flashcard 27: What is the impact of high leverage points in regression?

Answer: Influence regression line. Points far from mean $x$ have greater impact on line.

Flashcard 28: Identify the point where the regression line crosses the y-axis.

Answer: Y-intercept. The value of $y$ when $x = 0$ (the constant term).

Flashcard 29: What does a residual of 0 indicate?

Answer: Perfect prediction. Model predicted the exact observed value.

Flashcard 30: What is the impact of high leverage points in regression?

Answer: Influence regression line. Points far from mean $x$ have greater impact on line.

Flashcard 31: What does it mean if residuals show a pattern?

Answer: Model is not appropriate. Patterns suggest non-linear relationship or other violations.

Flashcard 32: Find the regression line if $b_0=2$ and $b_1=4$.

Answer: $y = 2 + 4x$. Substituting values into the standard form equation.

Flashcard 33: What does extrapolation mean in regression?

Answer: Predicting outside data range. Making predictions beyond the range of observed data.

Flashcard 34: Identify the point where the regression line crosses the y-axis.

Answer: Y-intercept. The value of $y$ when $x = 0$ (the constant term).

Flashcard 35: Identify the limitation of using $R^2$ alone to judge model fit.

Answer: Does not show causation. High $R^2$ doesn't prove the variables are causally related.

Flashcard 36: What does extrapolation mean in regression?

Answer: Predicting outside data range. Making predictions beyond the range of observed data.

Flashcard 37: State the independent variable in the equation $y = 5x + 3$.

Answer: $x$. The predictor or explanatory variable in the equation.

Flashcard 38: State the formula to calculate residuals.

Answer: $e_i = y_i - \bar{y}_i$. Difference between observed and predicted values.

Flashcard 39: What does a residual of 0 indicate?

Answer: Perfect prediction. Model predicted the exact observed value.

Flashcard 40: Find the predicted value of $y$ for $x=5$ in $y = 3x + 4$.

Answer:

1. Substitute $x=5$: $y = 3(5) + 4 = 19$.

Flashcard 41: What does a correlation coefficient of 0 imply?

Answer: No linear relationship. Zero correlation means no linear association between variables.

Flashcard 42: Determine the effect of adding a constant to all $y$ values.

Answer: Shifts intercept. Adding constant to all $y$ values shifts y-intercept up.

Flashcard 43: What does a positive slope indicate about the relationship between variables?

Answer: Positive correlation. Positive slope means $y$ increases as $x$ increases.

Flashcard 44: What does $R^2$ represent in a linear regression model?

Answer: Coefficient of determination. Measures proportion of variance in $y$ explained by the model.

Flashcard 45: Identify the y-intercept in $y = -2x + 6$.

Answer:

1. Y-intercept is the constant term when $x=0$.

Flashcard 46: State the condition when a linear model is appropriate.

Answer: Linear relationship. Data points should follow a straight-line pattern.

Flashcard 47: Find the correlation coefficient if $R^2 = 0.81$.

Answer: $\text{either } 0.9 \text{ or } -0.9$. $r = \pm\sqrt{R^2} = \pm\sqrt{0.81} = \pm 0.9$.

Flashcard 48: Find and correct: $y = 2x + 3$ is a nonlinear model.

Answer: Linear model. $y = 2x + 3$ is linear since it has constant slope.

Flashcard 49: Identify the term for unexplained variability in regression.

Answer: Error variance. Variability not explained by the regression model.

Flashcard 50: What is the significance of the $p$-value in regression?

Answer: Tests hypothesis about slope. Tests if slope is significantly different from zero.

Flashcard 51: What is the interpretation of a negative slope?

Answer: Negative correlation. Negative slope means $y$ decreases as $x$ increases.

Flashcard 52: What is the effect of an outlier on a regression line?

Answer: Influences the slope and intercept. Outliers can dramatically alter the regression line's position.

Flashcard 53: What is the purpose of a residual plot?

Answer: Assess model fit. Shows patterns that indicate model assumptions violations.

Flashcard 54: What does it mean if residuals show a pattern?

Answer: Model is not appropriate. Patterns suggest non-linear relationship or other violations.

Flashcard 55: Find and correct: $y = 2x + 3$ is a nonlinear model.

Answer: Linear model. $y = 2x + 3$ is linear since it has constant slope.

Flashcard 56: Identify the slope in the regression equation $y = 3 + 2x$.

Answer:

1. The slope is the coefficient of $x$ in the linear equation.

Flashcard 57: What does a $p$-value less than 0.05 indicate?

Answer: Statistically significant. Evidence to reject null hypothesis at 5% significance level.

Flashcard 58: Find the correlation coefficient if $R^2 = 0.81$.

Answer: $\text{either } 0.9 \text{ or } -0.9$. $r = \pm\sqrt{R^2} = \pm\sqrt{0.81} = \pm 0.9$.

Flashcard 59: What does $R^2$ represent in a linear regression model?

Answer: Coefficient of determination. Measures proportion of variance in $y$ explained by the model.

Flashcard 60: Choose the term for residuals in regression analysis.

Answer: Errors. Residuals are the differences between observed and predicted values.

Flashcard 61: Identify the dependent variable in regression analysis.

Answer: Response variable. The variable being predicted or explained ($y$).

Flashcard 62: Identify the dependent variable in regression analysis.

Answer: Response variable. The variable being predicted or explained ($y$).

Flashcard 63: Identify the slope in the regression equation $y = 3 + 2x$.

Answer:

1. The slope is the coefficient of $x$ in the linear equation.

Flashcard 64: What is the interpretation of a negative slope?

Answer: Negative correlation. Negative slope means $y$ decreases as $x$ increases.

Flashcard 65: Identify the term for unexplained variability in regression.

Answer: Error variance. Variability not explained by the regression model.

Flashcard 66: Find the $R^2$ if the correlation $r$ is 0.8.

Answer: 0.64. $R^2 = r^2$, so $(0.8)^2 = 0.64$.

Flashcard 67: What does a correlation coefficient of 0 imply?

Answer: No linear relationship. Zero correlation means no linear association between variables.