Linear Regression worksheet

1. Regression line: ŷ = 4.8462x + 47.1538

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.166667$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=62.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=10.833333$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=52.5$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{52.5}{10.833333}$$=4.846154$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=62.5-4.846154\times 3.166667$$=47.153846$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 4.8462x + 47.1538}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{52.5}{\sqrt{10.833333\times 337.5}}$$=0.868243$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.868243}^2$$=0.753846$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=4.846154\times 6+47.153846$$=76.230769$

2. Regression line: ŷ = 3.9714x + 44.6

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.5$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=58.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.5$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=69.5$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{69.5}{17.5}$$=3.971429$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=58.5-3.971429\times 3.5$$=44.6$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 3.9714x + 44.6}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{69.5}{\sqrt{17.5\times 729.5}}$$=0.61511$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.61511}^2$$=0.378361$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=3.971429\times 10+44.6$$=84.314286$

3. Regression line: ŷ = 4.3714x + 47.2

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.5$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=62.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.5$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=76.5$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{76.5}{17.5}$$=4.371429$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=62.5-4.371429\times 3.5$$=47.2$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 4.3714x + 47.2}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{76.5}{\sqrt{17.5\times 337.5}}$$=0.995418$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.995418}^2$$=0.990857$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=4.371429\times 6+47.2$$=73.428571$

4. Regression line: ŷ = 3.3429x + 54.4667

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.5$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=66.166667$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.5$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=58.5$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{58.5}{17.5}$$=3.342857$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=66.166667-3.342857\times 3.5$$=54.466667$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 3.3429x + 54.4667}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{58.5}{\sqrt{17.5\times 604.833333}}$$=0.568616$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.568616}^2$$=0.323324$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=3.342857\times 7+54.466667$$=77.866667$

5. Regression line: ŷ = 4.0286x + 48.4

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.5$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=62.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.5$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=70.5$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{70.5}{17.5}$$=4.028571$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=62.5-4.028571\times 3.5$$=48.4$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 4.0286x + 48.4}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{70.5}{\sqrt{17.5\times 337.5}}$$=0.917346$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.917346}^2$$=0.841524$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=4.028571\times 5+48.4$$=68.542857$

6. Regression line: ŷ = 4.05x + 49

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.333333$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=62.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=13.333333$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=54$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{54}{13.333333}$$=4.05$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=62.5-4.05\times 3.333333$$=49$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 4.05x + 49}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{54}{\sqrt{13.333333\times 337.5}}$$=0.804984$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.804984}^2$$=0.648$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=4.05\times 7+49$$=77.35$

7. Regression line: ŷ = 4.725x + 45.175

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.666667$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=62.5$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=13.333333$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=63$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{63}{13.333333}$$=4.725$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=62.5-4.725\times 3.666667$$=45.175$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 4.725x + 45.175}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{63}{\sqrt{13.333333\times 337.5}}$$=0.939149$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.939149}^2$$=0.882$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=4.725\times 6+45.175$$=73.525$

8. Regression line: ŷ = 6.0571x + 44.1333

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.5$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=65.333333$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.5$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=106$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{106}{17.5}$$=6.057143$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=65.333333-6.057143\times 3.5$$=44.133333$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 6.0571x + 44.1333}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{106}{\sqrt{17.5\times 939.333333}}$$=0.826755$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.826755}^2$$=0.683524$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=6.057143\times 5+44.133333$$=74.419048$

9. Regression line: ŷ = 7.9423x + 39.2115

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.666667$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=68.333333$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=17.333333$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=137.666667$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{137.666667}{17.333333}$$=7.942308$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=68.333333-7.942308\times 3.666667$$=39.211538$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 7.9423x + 39.2115}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{137.666667}{\sqrt{17.333333\times \mathrm{1,621.333333}}}$$=0.821205$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.821205}^2$$=0.674378$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=7.942308\times 5+39.211538$$=78.923077$

10. Regression line: ŷ = 2.2317x + 60.8171

1. Find the average (mean) of the x values, written x̄

   $\mathrm{x̄}$$=3.666667$

2. Find the average (mean) of the y values, written ȳ

   $ȳ$$=69$

3. Measure how spread out x is (Sxx): square each x value's distance from the mean, then add them up

   $\mathrm{Sxx}$$=27.333333$

4. Measure how x and y move together (Sxy): multiply each point's x and y distances from their means, then add them up

   $\mathrm{Sxy}$$=61$

5. Find the slope b, how much y changes for each step in x: divide Sxy by Sxx

   $b$$=\frac{61}{27.333333}$$=2.231707$

6. Find the intercept a, where the line crosses the y-axis. It puts the line through the point of the two averages

   $a$$=69-2.231707\times 3.666667$$=60.817073$

7. Put the slope and intercept into the line's equation

   $\text{ŷ = 2.2317x + 60.8171}$

8. Find the correlation r, how tightly the points follow a line. It compares Sxy with the spread of both x and y (Syy is the spread of y)

   $r$$=\frac{61}{\sqrt{27.333333\times 524}}$$=0.509704$

9. Square r to get R², the share of the ups and downs in y that the line explains

   $\mathrm{R²}$$={0.509704}^2$$=0.259798$

10. Put your new x into the equation to predict y (written ŷ)

    $ŷ$$=2.231707\times 6+60.817073$$=74.207317$