Exercises: Assess Fit by Analyzing Residuals

A function is fitted to data. For one data point, the observed value is $14$ and the model's predicted value is $17$. What is the residual at this point?

A line $y = 0.10x + 20$ models a phone bill, where $x$ is minutes used. At $x = 50$ minutes a customer was actually charged $\text{\char"0024}26$. The model predicts $0.10(50) + 20 = \text{\char"0024}25$. What is the residual (observed $-$ predicted) for this customer?

A student builds a residual plot and expects it to look just like the original scatter plot of the data. Why is that expectation wrong?

The scatter plot below shows data with a fitted line. For the highlighted point, the observed value is $30$ and the line predicts $34$ at that $x$. What is the residual (observed $-$ predicted)?

A line $\hat{y} = 2x + 5$ is fitted to data. At $x = 4$ the observed value is $11$. Compute the residual (observed $-$ predicted) at this point.

You have these residuals at the given $x$-values: $(1, +2)$, $(2, -1)$, $(3, +3)$, $(4, -2)$. To build the residual plot, where does the point for $x = 3$ go relative to the zero line?

The residual plot below shows residuals scattered randomly around the zero line with no clear pattern. What does this say about the fit of the model?

A residual plot shows a clear U-shaped pattern: the residuals are positive at the left and right ends and negative in the middle. What is the most likely problem, and what should you do?