Linear Regression — Practice Problems and Worked Solutions

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18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[1,2,3\right],y=\left[2,4,6\right]$$

   Answer: $\left[\begin{array}{c}0\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}0\\ 2\end{array}\right]$$
2. 2.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[1,2,3\right],y=\left[3,5,7\right]$$

   Answer: $\left[\begin{array}{c}1\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}1\\ 2\end{array}\right]$$
3. 3.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[0,1,2\right],y=\left[1,4,7\right]$$

   Answer: $\left[\begin{array}{c}1\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}1\\ 3\end{array}\right]$$
4. 4.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[1,3,5\right],y=\left[5,4,3\right]$$

   Answer: $\left[\begin{array}{c}\frac{11}{2}\\ -\frac{1}{2}\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}\frac{11}{2}\\ -\frac{1}{2}\end{array}\right]$$
5. 5.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[0,2,4\right],y=\left[2,3,4\right]$$

   Answer: $\left[\begin{array}{c}2\\ \frac{1}{2}\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}2\\ \frac{1}{2}\end{array}\right]$$
6. 6.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[-1,0,1\right],y=\left[4,2,0\right]$$

   Answer: $\left[\begin{array}{c}2\\ -2\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}2\\ -2\end{array}\right]$$
7. 7.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[1,2,3\right],y=\left[4,7,10\right]$$

   Answer: $\left[\begin{array}{c}1\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}1\\ 3\end{array}\right]$$
8. 8.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[0,1,2\right],y=\left[5,3,1\right]$$

   Answer: $\left[\begin{array}{c}5\\ -2\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}5\\ -2\end{array}\right]$$
9. 9.

   Find [intercept, slope] for the least-squares line y=a+bx.

   $$x=\left[2,4,6\right],y=\left[1,5,9\right]$$

   Answer: $\left[\begin{array}{c}-3\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
   2. $$a=\bar{y}-b\bar{x}$$
   3. $$[a,b]=\left[\begin{array}{c}-3\\ 2\end{array}\right]$$
10. 10.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[1,2,3\right],y=\left[1,3,5\right]$$

    Answer: $\left[\begin{array}{c}-1\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}-1\\ 2\end{array}\right]$$
11. 11.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[0,3,6\right],y=\left[2,4,6\right]$$

    Answer: $\left[\begin{array}{c}2\\ \frac{2}{3}\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}2\\ \frac{2}{3}\end{array}\right]$$
12. 12.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[-2,0,2\right],y=\left[1,2,3\right]$$

    Answer: $\left[\begin{array}{c}2\\ \frac{1}{2}\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}2\\ \frac{1}{2}\end{array}\right]$$
13. 13.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[1,4,7\right],y=\left[3,6,9\right]$$

    Answer: $\left[\begin{array}{c}2\\ 1\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}2\\ 1\end{array}\right]$$
14. 14.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[0,2,4\right],y=\left[7,5,3\right]$$

    Answer: $\left[\begin{array}{c}7\\ -1\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}7\\ -1\end{array}\right]$$
15. 15.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[1,2,3\right],y=\left[6,4,2\right]$$

    Answer: $\left[\begin{array}{c}8\\ -2\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}8\\ -2\end{array}\right]$$
16. 16.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[2,3,4\right],y=\left[5,8,11\right]$$

    Answer: $\left[\begin{array}{c}-1\\ 3\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}-1\\ 3\end{array}\right]$$
17. 17.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[0,1,2\right],y=\left[0,3,6\right]$$

    Answer: $\left[\begin{array}{c}0\\ 3\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}0\\ 3\end{array}\right]$$
18. 18.

    Find [intercept, slope] for the least-squares line y=a+bx.

    $$x=\left[1,3,5\right],y=\left[2,8,14\right]$$

    Answer: $\left[\begin{array}{c}-1\\ 3\end{array}\right]$

    Step-by-step solution

    1. $$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$
    2. $$a=\bar{y}-b\bar{x}$$
    3. $$[a,b]=\left[\begin{array}{c}-1\\ 3\end{array}\right]$$

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