Question Find the equation of the line of best fit for the data: (4, 3), (6, 4), (8, 9), (11, 12), (13, 17). Round the slope and y-intercept to 3 decimal places.

Studdy Solution

STEP 1

Assumptions
1. We are given a set of data points $(x, y)$ .
2. We need to find the equation of the line of best fit for the given data.
3. The line of best fit can be described by the equation $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
4. The slope $m$ can be calculated using the formula $m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ .
5. The y-intercept $b$ can be calculated using the formula $b = \frac{\sum y - m(\sum x)}{n}$ .
6. We will round the slope and y-intercept to three decimal places.

STEP 2

First, we will list the given data points for convenience:
$(x, y) = \{(4, 3), (6, 4), (8, 9), (11, 12), (13, 17)\}$

STEP 3

Calculate the sum of the $x$ values, $\sum x$ .
$\sum x = 4 + 6 + 8 + 11 + 13$

STEP 4

Calculate the sum of the $x$ values.
$\sum x = 4 + 6 + 8 + 11 + 13 = 42$

STEP 5

Calculate the sum of the $y$ values, $\sum y$ .
$\sum y = 3 + 4 + 9 + 12 + 17$

STEP 6

Calculate the sum of the $y$ values.
$\sum y = 3 + 4 + 9 + 12 + 17 = 45$

STEP 7

Calculate the sum of the product of $x$ and $y$ values, $\sum xy$ .
$\sum xy = (4 \cdot 3) + (6 \cdot 4) + (8 \cdot 9) + (11 \cdot 12) + (13 \cdot 17)$

STEP 8

Calculate the sum of the product of $x$ and $y$ values.
$\sum xy = (4 \cdot 3) + (6 \cdot 4) + (8 \cdot 9) + (11 \cdot 12) + (13 \cdot 17) = 12 + 24 + 72 + 132 + 221$

STEP 9

Calculate the sum of the product of $x$ and $y$ values.
$\sum xy = 12 + 24 + 72 + 132 + 221 = 461$

STEP 10

Calculate the sum of the squares of $x$ values, $\sum x^2$ .
$\sum x^2 = 4^2 + 6^2 + 8^2 + 11^2 + 13^2$

STEP 11

Calculate the sum of the squares of $x$ values.
$\sum x^2 = 4^2 + 6^2 + 8^2 + 11^2 + 13^2 = 16 + 36 + 64 + 121 + 169$

STEP 12

Calculate the sum of the squares of $x$ values.
$\sum x^2 = 16 + 36 + 64 + 121 + 169 = 406$

STEP 13

Calculate the number of data points, $n$ .
$n = 5$

STEP 14

Now we will calculate the slope $m$ of the line using the formula:
$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$

STEP 15

Plug in the values into the slope formula.
$m = \frac{5(461) - (42)(45)}{5(406) - (42)^2}$

STEP 16

Calculate the numerator and denominator of the slope formula.
$m = \frac{2305 - 1890}{2030 - 1764}$

STEP 17

Calculate the numerator and denominator of the slope formula.
$m = \frac{415}{266}$

STEP 18

Calculate the slope $m$ .
$m = \frac{415}{266} \approx 1.560$

STEP 19

Now we will calculate the y-intercept $b$ using the formula:
$b = \frac{\sum y - m(\sum x)}{n}$

STEP 20

Plug in the values into the y-intercept formula.
$b = \frac{45 - 1.560(42)}{5}$

STEP 21

Calculate the numerator of the y-intercept formula.
$b = \frac{45 - 65.52}{5}$

STEP 22

Calculate the numerator of the y-intercept formula.
$b = \frac{-20.52}{5}$

STEP 23

Calculate the y-intercept $b$ .
$b = \frac{-20.52}{5} \approx -4.104$

STEP 24

Now we have the slope $m$ and y-intercept $b$ . The equation of the line of best fit is:
$y = mx + b$

STEP 25

Plug in the values of $m$ and $b$ into the equation.
$y = 1.560x - 4.104$

STEP 26

Round the slope and y-intercept to three decimal places.
$y = 1.560x - 4.105$

STEP 27

Compare the final equation with the given options.
The correct equation of the line of best fit is:
$y = 1.560x - 4.105$
Therefore, the answer is A. $y=1.560x-4.105$ .

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