Back to Exercise: Assess fit by analyzing residuals

Exercises: Assess Fit by Analyzing Residuals

Work through each section in order. A residual is observed minus
predicted: positive when the data point is ABOVE the model, negative
when below. Keep the sign. When reading a residual plot, remember the
key idea: random scatter around zero means a GOOD fit, while a clear
pattern (a curve or a fan) means the model is wrong.

Grade 10·20 problems·~35 min·Common Core Math - HS Statistics and Probability·standard·hss-id-b-6b

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Warm-Up: What Is a Residual?

These problems review the residual definition and its sign.

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?

$-3$ , because residual = observed $-$ predicted $= 14 - 17$

$3$ , because a residual measures how far off the model is, which cannot be negative

$17 - 14 = 3$ , predicted minus observed

$14 + 17 = 31$

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 residual plot shows only the misses (residual vs $x$ ), centered on zero with the trend removed, so it is a different picture from the data cloud.

The residual plot is identical to the scatter plot, just rotated.

The residual plot always shows a straight increasing line.

A residual plot and a scatter plot are the same graph with different labels.

Fluency Practice

Compute residuals and read residual plots.

$A scatter plot with a fitted line; a dashed vertical segment marks the residual between one data point and the line.$

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?

At height $+3$ above the zero line, plotted at $x = 3$ .

At height $3$ below the zero line, plotted at $x = 3$ .

At the original observed $y$ -value, not at the residual.

On the zero line, since residuals always lie on zero.

$A residual plot with points scattered randomly above and below a dashed horizontal zero line, showing no pattern.$

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?

The model fits well; random scatter around zero means only noise is left over.

The model is poor, because the residuals are not all exactly zero.

The model is poor, because some residuals are negative.

You cannot tell anything from a residual plot.

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?

The data is curved and a line was the wrong family; refit with a quadratic or exponential.

The data was recorded incorrectly and should be thrown out.

Nothing is wrong; a U-shape is the normal look of a good fit.

The residuals should be made positive to remove the pattern.

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