Question

Find the slope intercept form of the equation of the line that best fits the data

Answer 1

The equation of the line that best fits the data, obtained using the least squares method is;
`\hat{y}` ≈ 0.03478·x + 5.7838

The steps used to find the best fit line equation is presented as follows;

The values in the table can be presented as follows;

X | Y

50 10

110 10

230 10

660 30

720 30

970 40

The least squares method equation can be used to find the slope-intercept of the equation of the line that best fits the data as follows;

The equation of the line in slope-intercept form, using the least squared method is;
`\hat{y}` = m·x + c

Where;

`m = \frac{\sum\limits_(i=1)^n(x_i-\bar{x})\cdot (y_i-\bar{y)}}{\sum\limits_(i=1)^n(x_i-\bar{x})^2}`

c =
`\bar{y}` - m·
`\bar{x}`

The data in the dable can be evaluated using MS Excel to get;

`\bar{x}` ≈ 456.6667

`\bar{y}` ≈ 21.66667

`\sum\limits_(i=1)^n(x_i-\bar{x})\cdot (y_i-\bar{y)}` = 24733.33

`{\sum\limits_(i=1)^n(x_i-\bar{x})^2}` = 711133.3

Therefore;
`m = \frac{\sum\limits_(i=1)^n(x_i-\bar{x})\cdot (y_i-\bar{y)}}{\sum\limits_(i=1)^n(x_i-\bar{x})^2}` = 24733.33/711133.3

24733.33/711133.3 ≈ 0.03478

m ≈ 0.03478

c ≈ 21.66667 - 0.03478 × 456.6667

c ≈ 5.7838

The equation of the best fit line is therefore;
`\hat{y}` ≈ 0.03478·x + 5.7838