Least Squares Regression

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#8049

Least Squares Line of Best Fit

$$y=mx+c$$ 

$$m=r\frac{s_y}{s_x}$$

$$y-intercept = \bar{y}-m\bar{x}$$

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#14939

The heights (x) and weights (y) of 11 people have been recorded, and the values of the following statistics determined: 

$$\bar{x}$$ = 173.2727cm,  s_x = 7.4443cm,  $$\bar{y}$$  = 6 5.4545cm,  s_y = 7.5943cm r = 0.8502.

Find the equation of the least squares regression line that will enable weight to be predicted from height.

y = mx + b

Find m using the formula: $$m=r\times\frac{s_y}{s_x}$$

$$m=0.8502\times\frac{7.5943}{7.443}$$

m = 0.867
Find b using the formula: $$\overline{y} -m\overline{x}$$

$$b=65.4545-0.867\times 173.2727$$

b = -84.773

y = 0.8671x – 84.773

[Admin]

#14941

Researchers have found a correlation between obesity and the number of hours per day children spend in front of a video screen.
Hours : $$\bar{H}=2.5,s_H=1.29$$         Kg over average weight: $$\bar{W}=2.17,s_W=1.05$$
Find the equation of the  least-squares line of best fit given the value of r = 0.955.

y = mx + b ⇔ W = mH + b

Find m using the formula: $$m=r\times\frac{s_y}{s_x}$$

$$m=0.995\times\frac{1.05}{1.29}$$

m = 0.81
Find b using the formula: $$\overline{y} -m\overline{x}$$

$$b=2.17-0.81\times 2.5$$

b = 0.145

W = 0.81H + 0.145

[Admin]

#18609

What is the y-intercept of the least squares regression line given m = 0.9, $$\bar{x}=40.10$$, and $$\bar{y}=60.44$$?

$$y-intercept = \bar{y}-m\bar{x}$$

y-intercept = 60.44 – 0.9 × 40.10

y-intercept = 24.35

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#18610

Researchers have found a correlation between the length of the legs of puppies and their running speed. Leg Length: $$\bar{x}=10.2,s_x=1.5$$         Running speed: $$\bar{y}=1.6,s_y=2.0$$ The least-squares line of best fit is drawn and the gradient of the line is 0.4. Find the value of r, the correlation coefficient.
$$0.4=r\frac{2.0}{1.5}$$	substitute the values into the equation: m = 0.4, sx = 1.5, sy = 2.0 and solve the equation for r
$$0.4=r\times\frac43$$	divide both sides by $$\frac43$$
$$r=0.4\div\frac43$$
r = 0.3

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#18612

Find the least squares line of best fit given the following data: $$\bar{x}=15.679,s_x=12.786$$      $$\bar{y}=85.128,s_y=5.985, r=0.865$$
$$m=0.865\times\frac{5.985}{12.786}$$ m = 0.405	Find m using the formula: $$m=r\times\frac{s_y}{s_x}$$
$$c=1.6-0.405\times 10.2$$ b = 78.778	Find c using the formula: $$\overline{y} -m\overline{x}$$
y = 0.405x + 78.778	least squares equation is in the form y = mx + c, so replace the values for m and c to make the equation

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