Question

identifying the correct residual plot
a scatterplot consists of (1, 4.0), (2, 3.3), (3, 3.8), (4, 2.6), and (5, 2.7). the line of best fit used to model the data is = -0.33x + 4.27. which residual plot is correct?

Explanation:

Step1: Recall Residual Formula

Residual \( r = y - \hat{y} \), where \( y \) is the actual value and \( \hat{y} \) is the predicted value from the line of best fit \( \hat{y} = -0.33x + 4.27 \).

Step2: Calculate Residual for \( x = 1 \), \( y = 4.0 \)

\( \hat{y} = -0.33(1) + 4.27 = 3.94 \)
\( r = 4.0 - 3.94 = 0.06 \) (close to 0, so point near x-axis at x=1)

Step3: Calculate Residual for \( x = 2 \), \( y = 3.3 \)

\( \hat{y} = -0.33(2) + 4.27 = 4.27 - 0.66 = 3.61 \)
\( r = 3.3 - 3.61 = -0.31 \) (negative, below x-axis at x=2)

Step4: Calculate Residual for \( x = 3 \), \( y = 3.8 \)

\( \hat{y} = -0.33(3) + 4.27 = 4.27 - 0.99 = 3.28 \)
\( r = 3.8 - 3.28 = 0.52 \) (positive, above x-axis at x=3)

Step5: Calculate Residual for \( x = 4 \), \( y = 2.6 \)

\( \hat{y} = -0.33(4) + 4.27 = 4.27 - 1.32 = 2.95 \)
\( r = 2.6 - 2.95 = -0.35 \) (negative, below x-axis at x=4)

Step6: Calculate Residual for \( x = 5 \), \( y = 2.7 \)

\( \hat{y} = -0.33(5) + 4.27 = 4.27 - 1.65 = 2.62 \)
\( r = 2.7 - 2.62 = 0.08 \) (close to 0, near x-axis at x=5)

Now, match these residuals (x=1: ~0, x=2: -0.31, x=3: +0.52, x=4: -0.35, x=5: ~0) to the plots. The second plot (from left) has: x=1 near 0, x=2 below, x=3 above, x=4 below, x=5 near 0, which matches.

Answer:

The second residual plot (from the left) is correct. (Assuming the options are labeled left to right as 1st, 2nd, 3rd, 4th; the 2nd plot matches the calculated residuals.)