To derive the linear regression equation that models the given data points [tex]\((65, 20), (68, 28), (73, 29), (77, 32), (78, 30)\)[/tex], we follow these steps:
1. Calculate the mean of the x-values (heights) and the y-values (points):
[tex]\[
\text{Mean of heights} = \frac{65 + 68 + 73 + 77 + 78}{5} = \frac{361}{5} = 72.2
\][/tex]
[tex]\[
\text{Mean of points} = \frac{20 + 28 + 29 + 32 + 30}{5} = \frac{139}{5} = 27.8
\][/tex]
2. Calculate the slope [tex]\(m\)[/tex] of the regression line:
The slope [tex]\(m\)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{\sum_{i=1}^n (x_i - \text{mean}_x) (y_i - \text{mean}_y)}{\sum_{i=1}^n (x_i - \text{mean}_x)^2}
\][/tex]
Here,
[tex]\[
\sum_{i=1}^n (x_i - \text{mean}_x) (y_i - \text{mean}_y) = (65 - 72.2)(20 - 27.8) + (68 - 72.2)(28 - 27.8) + (73 - 72.2)(29 - 27.8) + (77 - 72.2)(32 - 27.8) + (78 - 72.2)(30 - 27.8)
\][/tex]
[tex]\[
= (-7.2)(-7.8) + (-4.2)(0.2) + (0.8)(1.2) + (4.8)(4.2) + (5.8)(2.2)
\][/tex]
[tex]\[
= 56.16 + (-0.84) + 0.96 + 20.16 + 12.76 = 89.2
\][/tex]
[tex]\[
\sum_{i=1}^n (x_i - \text{mean}_x)^2 = (65 - 72.2)^2 + (68 - 72.2)^2 + (73 - 72.2)^2 + (77 - 72.2)^2 + (78 - 72.2)^2
\][/tex]
[tex]\[
= (-7.2)^2 + (-4.2)^2 + (0.8)^2 + (4.8)^2 + (5.8)^2
\][/tex]
[tex]\[
= 51.84 + 17.64 + 0.64 + 23.04 + 33.64 = 126.8
\][/tex]
Now, calculating the slope [tex]\(m\)[/tex]:
[tex]\[
m = \frac{89.2}{126.8} \approx 0.703
\][/tex]
3. Calculate the y-intercept [tex]\(b\)[/tex] of the regression line:
The intercept [tex]\(b\)[/tex] can be calculated using the formula:
[tex]\[
b = \text{mean}_y - m \times \text{mean}_x
\][/tex]
[tex]\[
b = 27.8 - 0.703 \times 72.2
\][/tex]
[tex]\[
b = 27.8 - 50.7906 \approx -22.991
\][/tex]
4. Formulate the linear regression equation:
With the slope [tex]\(m \approx 0.703\)[/tex] and y-intercept [tex]\(b \approx -22.991\)[/tex], the equation of the regression line is:
[tex]\[
y \approx 0.703 x - 22.991
\][/tex]
Therefore, the linear regression equation that models the data is:
[tex]\[
y \approx 0.703x - 22.991
\][/tex]
