Residuals — Quiz

Question 5 • Statistical Reasoning

The equation of the least-squares regression line is hat y is equal to 2 point 7 plus 1 point 0 4 2 x , where hat y is each son’s height and x is his father’s height. Which shows the residual plot?

Answer

Image with description A dot plot with the x axis titled Father's Height in Inches begins with 60 and ends at 76 with equal intervals of 2, such as 60, 62, 64, 66, 68, 70, 72, 74, and 76. The y axis titled Residual begins with negative 3 and ends at 5 with equal intervals of 1, such as negative 3, negative 1, negative 1, 0, 1, 2, 3, 4, and 5. A horizontal line parallel to the x axis starts from the point where y is 0. There is 1 dot at approximately 60, 0 point 8; 61, 0 point 6; 63, negative 1 point 2; 1 dot at 63, 2 point 8; 1 dot at 66, negative 1 point 5; 1 dot at 67, negative 2 point 5; 1 dot at 74, negative 2 point 8; and 1 dot at 75, 4 point 2.

Image with description A dot plot with the x axis titled Father's Height in Inches begins with 60 and ends at 76 with equal intervals of 2, such as 60, 62, 64, 66, 68, 70, 72, 74, and 76. The y axis titled Residual begins with negative 16 and ends at 4 with equal intervals of 4, such as negative 16, negative 12, negative 8, negative 4, 0, and 4. A horizontal line parallel to the x axis starts from the point where y is 0. There is a dot at approximately 60, 4; a dot at 61, 3; a dot at 63, negative 1; a dot at 63, 3; a dot at 66, 0; a dot at 67, 0; a dot at 74, negative 7; and a dot at 75, negative 15.

Image with description A dot plot with the x axis titled Father's Height in Inches begins with 60 and ends at 76 with equal intervals of 2, such as 60, 62, 64, 66, 68, 70, 72, 74, and 76. The y axis titled Residual begins with negative 5 and ends at 3 with equal intervals of 1, such as negative 5, negative 4, negative 3, negative 2, negative 1, 0, 1, 2, and 3. A horizontal line parallel to the x axis starts from the point where y is 0. There is a dot at approximately 60, negative 0 point 8; a dot at 61, negative 0 point 6; a dot at 63, negative 2 point 8; a dot at 63, 1 point 4; a dot at 66, negative 2 point 5; a dot at 67, 2 point 5; a dot at 74, 2 point 8; and a dot at 75, negative 4 point 2

Image with description A dot plot with the x axis titled Father's Height in Inches begins with 60 and ends at 76 with equal intervals of 2, such as 60, 62, 64, 66, 68, 70, 72, 74, and 76. The y axis titled Residual begins with negative 4 and ends at 16 with equal intervals of 4, such as negative 4, 0, 4, 8, 12, 16. A horizontal line parallel to the x axis starts from the point where y is 0. There is 1 dot at approximately 60, negative 4; 61, negative 3; 63, negative 3; 1 dot at 63, 1; 1 dot at 66, 0; 1 dot at 67, 0; 1 dot at 74, 7; and 1 dot at 75, 15.

Explanation

A residual is the observed son’s height minus the height predicted by the regression line: residual = y − (2.7 + 1.042x). For example, at a father’s height of 60 inches, the predicted height is 2.7 + 1.042(60) = 65.22 inches; an observed height of about 66 inches gives a residual of about 0.8 inches. The residuals should be plotted against the fathers’ heights, as in choice B.

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