Introducing Statistics: Are Variables Related - AP Statistics
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What does it mean if the residual plot shows a pattern?
What does it mean if the residual plot shows a pattern?
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Model may not be appropriate. Patterns suggest violations of regression assumptions.
Model may not be appropriate. Patterns suggest violations of regression assumptions.
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What is multicollinearity in the context of regression analysis?
What is multicollinearity in the context of regression analysis?
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When independent variables are highly correlated. Creates problems for interpreting individual variable effects.
When independent variables are highly correlated. Creates problems for interpreting individual variable effects.
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Find the correlation: All points on a scatter plot lie on a horizontal line.
Find the correlation: All points on a scatter plot lie on a horizontal line.
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No correlation. Horizontal line shows no relationship between variables.
No correlation. Horizontal line shows no relationship between variables.
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What does a low $p$-value indicate in a test for correlation?
What does a low $p$-value indicate in a test for correlation?
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Strong evidence against the null hypothesis. More likely to reject null of no correlation.
Strong evidence against the null hypothesis. More likely to reject null of no correlation.
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What is the significance of the intercept in a regression equation?
What is the significance of the intercept in a regression equation?
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The expected value of the dependent variable when the independent variable is zero. Starting value when predictor equals zero.
The expected value of the dependent variable when the independent variable is zero. Starting value when predictor equals zero.
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Find the correlation coefficient range for a strong negative relationship.
Find the correlation coefficient range for a strong negative relationship.
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Between -1 and -0.7. Values near -1 indicate strong negative association.
Between -1 and -0.7. Values near -1 indicate strong negative association.
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State the formula for calculating the correlation coefficient $r$.
State the formula for calculating the correlation coefficient $r$.
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$r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}}$. Standardized covariance divided by product of standard deviations.
$r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}}$. Standardized covariance divided by product of standard deviations.
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What is the meaning of a correlation coefficient of -1?
What is the meaning of a correlation coefficient of -1?
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Perfect negative linear relationship. All points lie on a perfectly downward-sloping line.
Perfect negative linear relationship. All points lie on a perfectly downward-sloping line.
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What is the meaning of a correlation coefficient of 1?
What is the meaning of a correlation coefficient of 1?
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Perfect positive linear relationship. All points lie on a perfectly upward-sloping line.
Perfect positive linear relationship. All points lie on a perfectly upward-sloping line.
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Define causal relationship in statistics.
Define causal relationship in statistics.
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A relationship where one variable directly affects another. Changes in one variable cause changes in another.
A relationship where one variable directly affects another. Changes in one variable cause changes in another.
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What type of plot is typically used to visualize relationships between two variables?
What type of plot is typically used to visualize relationships between two variables?
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Scatter plot. Shows the association pattern between two quantitative variables.
Scatter plot. Shows the association pattern between two quantitative variables.
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Identify the correlation: Scatter plot points lie close to an upward-sloping line.
Identify the correlation: Scatter plot points lie close to an upward-sloping line.
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Strong positive correlation. Points cluster tightly around an increasing trend.
Strong positive correlation. Points cluster tightly around an increasing trend.
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Identify the correlation: Scatter plot points are widely scattered with no pattern.
Identify the correlation: Scatter plot points are widely scattered with no pattern.
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No correlation. Points show no discernible linear pattern.
No correlation. Points show no discernible linear pattern.
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What does the slope of a regression line indicate?
What does the slope of a regression line indicate?
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The rate of change of the dependent variable per unit change in the independent variable. Shows how much $y$ changes per unit increase in $x$.
The rate of change of the dependent variable per unit change in the independent variable. Shows how much $y$ changes per unit increase in $x$.
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Which statistical test is used to determine if two variables are independent?
Which statistical test is used to determine if two variables are independent?
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Chi-square test. Tests whether categorical variables are related.
Chi-square test. Tests whether categorical variables are related.
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What is the null hypothesis in a test for independence between two variables?
What is the null hypothesis in a test for independence between two variables?
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The variables are independent. Assumes no association between the variables.
The variables are independent. Assumes no association between the variables.
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What is the alternative hypothesis in a test for independence between two variables?
What is the alternative hypothesis in a test for independence between two variables?
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The variables are not independent. Claims there is an association between variables.
The variables are not independent. Claims there is an association between variables.
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State the formula for the chi-square statistic.
State the formula for the chi-square statistic.
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$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$. Compares observed and expected frequencies.
$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$. Compares observed and expected frequencies.
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Which coefficient measures the strength and direction of a linear relationship?
Which coefficient measures the strength and direction of a linear relationship?
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Pearson correlation coefficient. Quantifies linear association strength and direction.
Pearson correlation coefficient. Quantifies linear association strength and direction.
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What is the Spearman rank correlation coefficient used for?
What is the Spearman rank correlation coefficient used for?
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Measuring the strength and direction of a monotonic relationship. Works with ranked data instead of raw values.
Measuring the strength and direction of a monotonic relationship. Works with ranked data instead of raw values.
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What does a high $p$-value indicate in a test for correlation?
What does a high $p$-value indicate in a test for correlation?
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Weak evidence against the null hypothesis. Less likely to reject null of no correlation.
Weak evidence against the null hypothesis. Less likely to reject null of no correlation.
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What is the term for variables that show a relationship in a scatter plot?
What is the term for variables that show a relationship in a scatter plot?
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Correlated variables. Variables that show association in their values.
Correlated variables. Variables that show association in their values.
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What statistical method predicts the value of a dependent variable?
What statistical method predicts the value of a dependent variable?
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Regression analysis. Uses relationships to make predictions about outcomes.
Regression analysis. Uses relationships to make predictions about outcomes.
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Identify the error: Assuming correlation implies causation.
Identify the error: Assuming correlation implies causation.
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Correlation does not imply causation. Association doesn't prove one variable causes another.
Correlation does not imply causation. Association doesn't prove one variable causes another.
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What is the meaning of a residual in regression analysis?
What is the meaning of a residual in regression analysis?
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The difference between observed and predicted values. Measures prediction error for each data point.
The difference between observed and predicted values. Measures prediction error for each data point.
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What type of plot helps detect non-linearity in relationships?
What type of plot helps detect non-linearity in relationships?
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Residual plot. Patterns in residuals reveal model inadequacies.
Residual plot. Patterns in residuals reveal model inadequacies.
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Which variable is plotted on the x-axis in a scatter plot?
Which variable is plotted on the x-axis in a scatter plot?
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Independent variable. By convention, the explanatory variable goes on x-axis.
Independent variable. By convention, the explanatory variable goes on x-axis.
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Find the $R^2$ value indicating perfect explanatory power.
Find the $R^2$ value indicating perfect explanatory power.
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$R^2 = 1$. Model explains all the variability.
$R^2 = 1$. Model explains all the variability.
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What does the slope of a regression line indicate?
What does the slope of a regression line indicate?
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The rate of change of the dependent variable per unit change in the independent variable. Shows how much $y$ changes per unit increase in $x$.
The rate of change of the dependent variable per unit change in the independent variable. Shows how much $y$ changes per unit increase in $x$.
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Which term describes a correlation due to a third variable?
Which term describes a correlation due to a third variable?
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Spurious correlation. False correlation caused by confounding variable.
Spurious correlation. False correlation caused by confounding variable.
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