Worked solution P33-Easy-1. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-2. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-3. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-4. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-5. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-6. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-7. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-8. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-9. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-10. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-11. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-12. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-13. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-14. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-15. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Easy-16. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-1. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-2. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-3. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-4. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-5. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-6. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-7. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-8. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-9. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-10. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-11. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-12. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-13. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-14. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-15. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Tough-16. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-1. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-2. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-3. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-4. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-5. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-6. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-7. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-8. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-9. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-10. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-11. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-12. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-13. The least-squares line is approximately $\widehat{y}=5.267+3.971x$, with $r=0.981$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=3.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 3.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-14. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-15. The least-squares line is approximately $\widehat{y}=5.267+2.971x$, with $r=0.967$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=2.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 2.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.

Worked solution P33-Toughest-16. The least-squares line is approximately $\widehat{y}=5.267+1.971x$, with $r=0.929$. At x=4, the residual is 1.848. $$b=r\frac{s_y}{s_x}=1.971,a=\overline{y}-b\overline{x}=5.267,e=y-\widehat{y}=1.848.$$ Interpretation: Within the observed range, each one-unit increase in x is associated with a predicted increase of about 1.971 y-units. Validity: Inspect form, unusual points, and residual pattern before relying on a linear model or extrapolating. Error to reject: Correlation and slope describe association; neither proves that changing x causes y to change.