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What are the differences between a good residual plot and a bad residual plot?
Good: Random scatter, no pattern | Bad: Clear pattern (curve, funnel), indicates non-linearity.
What are the differences between a positive residual and a negative residual?
Positive: Model underestimated, y > ŷ | Negative: Model overestimated, y < ŷ
What are the key differences between a scatterplot and a residual plot?
Scatterplot: Shows relationship between x and y | Residual Plot: Shows residuals (y-ŷ) against x, assessing model fit.
Compare the implications of residuals clustered near zero versus residuals spread far from zero.
Near Zero: Model predictions are close to actual values, better fit | Far From Zero: Model predictions are less accurate, poorer fit.
What are the differences between using a linear model and a non-linear model based on residual plots?
Linear Model: Appropriate when residuals are randomly scattered | Non-Linear Model: More appropriate when residuals show a pattern.
What is the formula for calculating a residual?
$$Residual = y - \hat{y}$$
What is the goal of a linear regression model in terms of residuals?
Minimize the sum of the squared residuals (least squares criterion).
How do you calculate the predicted value (ŷ) using the Least Squares Regression Line (LSRL)?
Using the equation of the LSRL: ŷ = a + bx, where a is the y-intercept and b is the slope.
Given the LSRL and a data point (x, y), how do you calculate the residual?
1. Find the predicted value (ŷ) using the LSRL. 2. Subtract the predicted value from the actual value (y): Residual = y - ŷ.
What does a residual of 0 indicate?
The model perfectly predicted the observed value.
Explain the concept of a residual plot and its purpose.
A residual plot is a graph that displays residuals on the y-axis and predictor variable on the x-axis. It helps assess the appropriateness of a linear model.
Explain the concept of 'least squares criterion'.
The 'least squares criterion' aims to minimize the sum of the squares of the residuals, ensuring the best fit for the regression line.
Explain what a pattern in a residual plot suggests.
A pattern (e.g., curve, funnel) suggests that a linear model is not appropriate for the data and a non-linear model might be a better fit.
Explain what a random scatter in a residual plot suggests.
A random scatter suggests that a linear model is a good fit for the data, as there's no systematic pattern in the residuals.
What does it mean if a linear model 'overestimates' a value?
The predicted value (ŷ) from the model is higher than the actual observed value (y). This results in a negative residual.
What does it mean if a linear model 'underestimates' a value?
The predicted value (ŷ) from the model is lower than the actual observed value (y). This results in a positive residual.