Question

What's the regression line and write an equation in y=ax+b form. Round a and b to the nearest tenth.

Answer 1

The data is plotted in the scatter graph shown below. The regression line shows a positive correlation. The equation of a straight line could normally be formed by reading the y-intercept and working out gradient, but in this graph, it would be tricky to read the y-intercept, so we will have to use the Least Square method.

We need the value of the slope (m) and the value of y-intercept (b). The formula to find both values are shown in picture 2 and picture 3 below
---------------------------------------------------------------------------------------------------------------
These following steps are for finding the slope

STEP 1: Find the mean of height and the mean of weight

Mean of height = ∑height ÷ number of data
Mean of height = $\frac{61+61+62+64+65+68+66+67+69+70}{10}=65.3$

Mean of weight = ∑weight ÷ number of data
Mean of weight = $\frac{99+104+110+133+130+142+146+153+184+185}{10}=124$

STEP 2: Subtract each value of height by its mean and subtract each value of weight by its mean

Weight                                                           Height
99 - 124 = -25                                         61 - 65.3 = -4.3
104 - 124 = -20                                        61 - 65.3 = -4.3
110 - 124 = -14                                          62 - 65.3 = -3.3
133 - 124 = 9                                           64 - 65.3 = -1.3
130 - 124 = 6                                           65 - 65.3 = -0.3
142 - 124 = 18                                          68 - 65.3 = 2.7
146 - 124 = 22                                         66 - 65.3 = 0.7
153 - 124 = 29                                         67 - 65.3 = 1.7
184 - 124 = 60                                         69 - 65.3 = 3.7
185 - 124 = 61                                          70 - 65.3 = 4.7

STEP 3: Multiply each pair of weight and height from STEP 2 and total the answers

(-25×-4.3) + (-20×-4.3) + (-14×-3.3) + (9×-1.3) + (6×-0.3) + (18×2.7) + (22×0.7) + (29×1.7) + (60×3.7) + (61×4.7)

107.5 + 86 + 46.2 + 11.7 + 1.8 + 48.6 + 15.4 + 49.3 + 222 + 286.7 = 875.2

STEP 4: Find (Value of weight - mean of weight)²
(-25)² + (-20)² + (-14)² + (9)² + (6)² + (18)² + (22)² + (29)² + (60)² + (61)² = 10308

The value of m = STEP 3÷STEP 4 = 875.2 ÷ 10308 = 0.085

----------------------------------------------------------------------------------------------------------------
To find the y-intercept, refer to formula in picture 3
We have,
The mean of weight (x) = 124
The mean of height (y) = 65.3
The slope (m) = 0.104

b = 65.3 - (0.085)(124) = 54.76

--------------------------------------------------------------------------------------------------------------

The equation of line best fit is $y=0.085x+5476$

---------------------------------------------------------------------------------------------------------------

To find coefficient of correlation, we are going to need these following values
A ∑(Weight × Height) ⇒ Multiply each pair of weight and height and total the numbers
B ∑(Weight - mean of weight)² ⇒ Subtract each weight by the mean of weight, square each answer then add up
C ∑(Height - mean of height)² ⇒ Subtract each height by the mean of height, square each answer then add up

After calculation, we have
∑(Weight  - Mean of weight) × (Height - Mean of height) = 875.2
∑(Weight - Mean of weight)² = 10308
∑(Height - Mean of height)² = 96.1

Coefficient of correlation is given by
$r= \frac{875.2}{ \sqrt{10308*96.1} }= 0.879$

The value 0.879 shows a strong positive correlation