Question

What is the equation of the line of best fit for the following data? Round the slope and y-intercept of the line to three decimal places. x y 2 2 5 8 7 10 9 11 11 13

Answer 1

SOLUTION

TO DETERMINE

The equation of the line of best fit for the following data ( Round the slope and y-intercept of the line to three decimal places. )

$\sf{x \: : \: \: 2 \: \: \: \: \: 5 \: \: \: \: 7 \: \: \: \: 9 \: \: \: \: 11 }$

$\sf{y \: : \: \: 2 \: \: \: \: \: 8\: \: \: \: 10 \: \: \: \: 11 \: \: \: \: 13 }$

EVALUATION

Let the equation of the line of best fit for the given data is

$\sf{y = a + bx \: \: \: \: \: - - - - (1)}$

Where a and b are given by

$\sf{ \sum \: y = 5a + b \sum \: x \: \: \: \: \: - - - - (2)}$

$\sf{ \sum \: xy = a \sum \: x + b \sum {x}^{2} \: \: \: \: \: - - - - (3)}$

We now from the table as below

$\begin{gathered} \begin{array}{|c|c| c|c| } \underline{\sf{x}} & \underline{\sf{y}}& \underline{\sf{ {x}^{2} }}& \underline{\sf{xy}} \\ \\ 2 & 2 & 4 & 4 \\5 & 8 & 25 & 40 \\7 & 10 & 40 & 70\\9 & 11 & 81 & 99\\11 & 13 & 121 & 143 \\ \\ \sf{\sum{x} = 34} & \sf{\sum{y} = 44} & \sf{\sum{ {x}^{2} } = 280} & \sf{\sum{xy} = 356} \end{array}\end{gathered}$

From Equation 2 we get

$\sf{5a + 34b = 44 \: \: \: - - - (4)}$

$\sf{34a + 280b = 356 \: \: \: - - - (5)}$

34 × Equation 4 - 5 × Equation 5 gives

$\sf{ - 244b = - 284 }$

$\sf{ \implies \: 244b = 284}$

$\sf{ \implies \: b = 1.1639}$

$\sf{ \implies \: b = 1.164}$

From Equation 4 we get

$\sf{ 5a = 4.424}$

$\sf{ \implies \: a = 0.8848}$

$\sf{ \implies \: a = 0.885}$

Hence the required equation of the line of best fit for the given data is

$\boxed{ \: \: \: \sf{y = 0.885 + 1.164x} \: \: \: } \:$

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