Linear regression, correlation coefficients, and residual analysis concepts for a college intro statistics course.
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- What is the Pearson correlation coefficient (r)?
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A measure of the strength and direction of the linear relationship between two variables, ranging from -1 to 1.- If the Pearson correlation coefficient is r = 0.85, what does this indicate?
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A strong positive linear relationship between the two variables.- What is the range of possible values for the Pearson correlation coefficient?
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From -1 to 1, inclusive.- What does r = 0 indicate about the relationship between two variables?
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No linear relationship between the two variables (though a non-linear relationship may exist).- Can a correlation of r = 0.9 prove that one variable causes changes in another?
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No, correlation does not imply causation; other factors or confounding variables may explain the relationship.- What is the primary goal of linear regression?
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To model and predict the value of a dependent variable based on one or more independent variables using a linear equation.- In the regression equation y = a + bx, what do a and b represent?
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a is the y-intercept and b is the slope of the regression line.- What does the slope (b) in a linear regression model tell us?
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The average change in the dependent variable for each one-unit increase in the independent variable.- What does the y-intercept (a) in a linear regression model represent?
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The predicted value of y when x = 0.- If a regression slope is b = 3.5, how do you interpret this value?
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For every one-unit increase in the independent variable, the dependent variable is predicted to increase by 3.5 units on average.- What is the least squares method in linear regression?
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A method that minimizes the sum of the squared residuals to find the best-fitting line through the data.- What is a residual in regression?
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The difference between an observed value and the predicted value from the regression model (residual = observed - predicted).- If a data point has an observed value of 45 and the regression model predicts 42, what is the residual?
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3 (calculated as 45 - 42).- What is the coefficient of determination (R-squared)?
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A measure of the proportion of variance in the dependent variable that is explained by the independent variable(s), ranging from 0 to 1.- If R-squared = 0.64, what percentage of the variance in the dependent variable is explained by the model?
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64%, meaning 36% of the variance remains unexplained.
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