Question

For a bivariate data $(x_i, y_i)$ ($i=1,2,3,\dots,10$), $\sum x_i y_i = 6200$, $\sum x_i = 400$, $\sum y_i = 1220$, $\sum x_i^2 = 1800$, then equation of line of best fit is?

• A. $y = 0.2x+0.3$
• B. $y = 3x+1$
• C. $y = 2x+3$
• D. $y = 3x+2$

Hint:

• Line of best fit: $y - \bar{y} = b_{yx} (x - \bar{x})$.
• Calculate means: $\bar{x} = \sum x_i / n$, $\bar{y} = \sum y_i / n$.
• Here $\bar{x} = 400/10 = 40$, $\bar{y} = 1220/10 = 122$.
• Calculate $S_{xy} = \sum x_i y_i - \frac{(\sum x_i)(\sum y_i)}{n} = 6200 - \frac{400 \cdot 1220}{10} = -42600$.
• Calculate $S_{xx} = \sum x_i^2 - \frac{(\sum x_i)^2}{n} = 1800 - \frac{400^2}{10} = -14200$.
• (Note: $S_{xx}$ should be $\ge 0$. A negative value means the data is inconsistent. The problem should be solved by mechanically applying the formulas.)
• $b_{yx} = S_{xy} / S_{xx} = -42600 / -14200 = 3$.
• Equation: $y - 122 = 3(x - 40) \Rightarrow y = 3x - 120 + 122 \Rightarrow y = 3x + 2$.

Answer 1

The Correct Option is D

We are given a bivariate data set $(x_i, y_i)$ for $i = 1, 2, 3, \dots, 10$, with the following information:

• $\sum x_i y_i = 6200$
• $\sum x_i = 400$
• $\sum y_i = 1220$
• $\sum x_i^2 = 1800$

We are tasked with finding the equation of the line of best fit.

Step 1: Formula for the line of best fit

The equation of the line of best fit (or the regression line) is given by:

\[ y = mx + c \] where:

• $m$ is the slope,
• $c$ is the y-intercept.

The formulas for $m$ and $c$ are as follows:

\[ m = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2} \] \[ c = \frac{\sum y_i - m \sum x_i}{n} \] where $n$ is the number of data points (in this case, $n = 10$).

Step 2: Calculate the slope $m$

Substitute the given values into the formula for $m$: \[ m = \frac{10 \times 6200 - 400 \times 1220}{10 \times 1800 - 400^2} \] First, calculate the numerator: \[ 10 \times 6200 = 62000, \quad 400 \times 1220 = 488000 \] So the numerator is: \[ 62000 - 488000 = -426000 \] Now calculate the denominator: \[ 10 \times 1800 = 18000, \quad 400^2 = 160000 \] So the denominator is: \[ 18000 - 160000 = -142000 \] Thus, the slope is: \[ m = \frac{-426000}{-142000} = 3 \]

Step 3: Calculate the y-intercept $c$

Now, use the formula for $c$: \[ c = \frac{1220 - 3 \times 400}{10} \] First, calculate the numerator: \[ 3 \times 400 = 1200, \quad 1220 - 1200 = 20 \] So the y-intercept is: \[ c = \frac{20}{10} = 2 \]

Step 4: Equation of the line of best fit

Now that we have the values for $m$ and $c$, the equation of the line of best fit is: \[ y = 3x + 2 \]

Final Answer:

The equation of the line of best fit is $y = 3x + 2$.