Representing Relationships Between Two Quantitative Variables - AP Statistics
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Define extrapolation in the context of regression.
Define extrapolation in the context of regression.
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Predicting outside the range of the data. Risky because patterns may not continue beyond data.
Predicting outside the range of the data. Risky because patterns may not continue beyond data.
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What is a scatterplot used for in statistics?
What is a scatterplot used for in statistics?
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Visualize relationship between two quantitative variables. Shows pattern and strength of association graphically.
Visualize relationship between two quantitative variables. Shows pattern and strength of association graphically.
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Describe the meaning of a residual in regression analysis.
Describe the meaning of a residual in regression analysis.
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Difference between observed and predicted values. Measures prediction error for each observation.
Difference between observed and predicted values. Measures prediction error for each observation.
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What does the coefficient of determination $R^2$ represent?
What does the coefficient of determination $R^2$ represent?
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Proportion of variance in $y$ explained by $x$. Indicates how much variation the model explains.
Proportion of variance in $y$ explained by $x$. Indicates how much variation the model explains.
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Calculate residual for $y = 15$, $\bar{y} = 10$, predicted $y = 13$.
Calculate residual for $y = 15$, $\bar{y} = 10$, predicted $y = 13$.
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Residual = 2. Observed minus predicted gives prediction error.
Residual = 2. Observed minus predicted gives prediction error.
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Identify the coefficient of determination and its symbol.
Identify the coefficient of determination and its symbol.
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$R^2$. Measures goodness of fit in regression models.
$R^2$. Measures goodness of fit in regression models.
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What is the role of the $y$-intercept $a$ in a regression line?
What is the role of the $y$-intercept $a$ in a regression line?
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Value of $y$ when $x = 0$. Starting value when explanatory variable equals zero.
Value of $y$ when $x = 0$. Starting value when explanatory variable equals zero.
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Determine the predicted value for $x = 3$ in $\hat{y} = 4 + 2x$.
Determine the predicted value for $x = 3$ in $\hat{y} = 4 + 2x$.
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Predicted $y = 10$. Substitute $x$ value into regression equation.
Predicted $y = 10$. Substitute $x$ value into regression equation.
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How do you interpret the slope $b$ in a regression line?
How do you interpret the slope $b$ in a regression line?
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Change in $y$ for a one-unit increase in $x$. Represents rate of change in response variable.
Change in $y$ for a one-unit increase in $x$. Represents rate of change in response variable.
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What is the least squares line for $b = 1.5$, $a = 2$?
What is the least squares line for $b = 1.5$, $a = 2$?
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$\hat{y} = 2 + 1.5x$. Standard linear equation form with given parameters.
$\hat{y} = 2 + 1.5x$. Standard linear equation form with given parameters.
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Determine the slope $b$ given $r = 0.5$, $s_y = 2$, $s_x = 4$.
Determine the slope $b$ given $r = 0.5$, $s_y = 2$, $s_x = 4$.
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$b = 0.25$. Formula: $b = r \cdot \frac{s_y}{s_x}$ gives slope.
$b = 0.25$. Formula: $b = r \cdot \frac{s_y}{s_x}$ gives slope.
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Find the residual for $y = 10$, $\bar{y} = 8$, predicted $y = 9$.
Find the residual for $y = 10$, $\bar{y} = 8$, predicted $y = 9$.
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Residual = $10 - 9 = 1$. Actual value minus predicted value gives error.
Residual = $10 - 9 = 1$. Actual value minus predicted value gives error.
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What does a correlation coefficient $r$ of 0 indicate?
What does a correlation coefficient $r$ of 0 indicate?
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No linear relationship. Zero correlation means no linear pattern exists.
No linear relationship. Zero correlation means no linear pattern exists.
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What is the purpose of transformation in regression?
What is the purpose of transformation in regression?
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Stabilize variance and make data normal. Improves model fit by meeting assumptions.
Stabilize variance and make data normal. Improves model fit by meeting assumptions.
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What is a leverage point in regression analysis?
What is a leverage point in regression analysis?
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Point with extreme value in predictor variable. High leverage can strongly influence regression results.
Point with extreme value in predictor variable. High leverage can strongly influence regression results.
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What does a residual plot help to assess?
What does a residual plot help to assess?
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Linearity and equal variance in regression. Random scatter indicates model assumptions are met.
Linearity and equal variance in regression. Random scatter indicates model assumptions are met.
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Determine if $r = -1.0$ indicates a perfect relationship.
Determine if $r = -1.0$ indicates a perfect relationship.
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Yes, perfect negative. Extreme values indicate perfect linear relationships.
Yes, perfect negative. Extreme values indicate perfect linear relationships.
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Determine the correlation sign: $r = -0.7$.
Determine the correlation sign: $r = -0.7$.
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Negative. Negative values indicate downward trend.
Negative. Negative values indicate downward trend.
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What is the correlation for no linear relationship?
What is the correlation for no linear relationship?
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$r = 0$. Zero indicates absence of linear relationship.
$r = 0$. Zero indicates absence of linear relationship.
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Find the residual for $y = 12$, predicted $y = 10$.
Find the residual for $y = 12$, predicted $y = 10$.
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Residual = 2. Simple subtraction: observed minus predicted.
Residual = 2. Simple subtraction: observed minus predicted.
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Identify the correct interpretation: $R^2 = 0.8$.
Identify the correct interpretation: $R^2 = 0.8$.
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80% of variance explained by model. Proportion of total variation explained by regression.
80% of variance explained by model. Proportion of total variation explained by regression.
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Find $R^2$ if $r = -0.5$.
Find $R^2$ if $r = -0.5$.
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$R^2 = 0.25$. Sign doesn't affect $R^2$ calculation.
$R^2 = 0.25$. Sign doesn't affect $R^2$ calculation.
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Find and correct: $r = 1.2$ is possible for correlation.
Find and correct: $r = 1.2$ is possible for correlation.
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Correct: $r$ must be $-1 \leq r \leq 1$. Correlation cannot exceed absolute value of 1.
Correct: $r$ must be $-1 \leq r \leq 1$. Correlation cannot exceed absolute value of 1.
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Determine the direction of association: $r = 0.6$.
Determine the direction of association: $r = 0.6$.
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Positive. Positive correlation indicates upward trend.
Positive. Positive correlation indicates upward trend.
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Identify the correlation strength: $r = -0.9$.
Identify the correlation strength: $r = -0.9$.
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Strong negative. Values near -1 or 1 indicate strong association.
Strong negative. Values near -1 or 1 indicate strong association.
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State the interpretation of a correlation coefficient $r$ of 0.8.
State the interpretation of a correlation coefficient $r$ of 0.8.
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Strong positive linear relationship. Values near 1 indicate strong positive linear association.
Strong positive linear relationship. Values near 1 indicate strong positive linear association.
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Identify the term for a point that significantly affects a regression line.
Identify the term for a point that significantly affects a regression line.
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Influential point. Has large effect on regression line parameters.
Influential point. Has large effect on regression line parameters.
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Find the $y$-intercept for $\hat{y} = 3 + 4x$.
Find the $y$-intercept for $\hat{y} = 3 + 4x$.
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$y$-intercept = 3. Constant term is value when $x = 0$.
$y$-intercept = 3. Constant term is value when $x = 0$.
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Identify the range of values for the correlation coefficient $r$.
Identify the range of values for the correlation coefficient $r$.
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$-1 \leq r \leq 1$. Correlation is bounded between perfect negative and perfect positive.
$-1 \leq r \leq 1$. Correlation is bounded between perfect negative and perfect positive.
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Calculate $R^2$ given $r = 0.6$.
Calculate $R^2$ given $r = 0.6$.
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$R^2 = 0.36$. Coefficient of determination equals correlation squared.
$R^2 = 0.36$. Coefficient of determination equals correlation squared.
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