To find the equation of the line of best fit, we need to calculate the slope and the $y$-intercept.
First, let's find the mean of $x$ and $y$.
Mean of $x$:
$(7 + 8 + 10 + 13 + 15)/5 = 53/5 = 10.6$
Mean of $y$:
$(25 + 29 + 41 + 48 + 57)/5 = 200/5 = 40$
Next, let's calculate the differences between each $x$ value and the mean of $x$, as well as the differences between each $y$ value and the mean of $y$.
For $x$:
$x_1 - \text{mean of } x = 7 - 10.6 = -3.6$
$x_2 - \text{mean of } x = 8 - 10.6 = -2.6$
$x_3 - \text{mean of } x = 10 - 10.6 = -0.6$
$x_4 - \text{mean of } x = 13 - 10.6 = 2.4$
$x_5 - \text{mean of } x = 15 - 10.6 = 4.4$
For $y$:
$y_1 - \text{mean of } y = 25 - 40 = -15$
$y_2 - \text{mean of } y = 29 - 40 = -11$
$y_3 - \text{mean of } y = 41 - 40 = 1$
$y_4 - \text{mean of } y = 48 - 40 = 8$
$y_5 - \text{mean of } y = 57 - 40 = 17$
Now, let's calculate the sum of:
- the product of the differences between $x_i$ and the mean of $x$, and
- the differences between $y_i$ and the mean of $y$.
$(-3.6 \cdot -15) + (-2.6 \cdot -11) + (-0.6 \cdot 1) + (2.4 \cdot 8) + (4.4 \cdot 17)$
$= 54 + 28.6 + -0.6 + 19.2 + 74.8$
$= 176$
Next, we'll calculate the sum of the squared differences for $x$ and divide it by $n-1$, where $n$ is the number of observations.
$( (-3.6)^2 + (-2.6)^2 + (-0.6)^2 + (2.4)^2 + (4.4)^2 )/4$
$= (12.96 + 6.76 + 0.36 + 5.76 + 19.36)/4$
$= (45.2)/4$
$= 11.3$
Finally, let's calculate the slope of the line.
Slope $ = \frac{\text{Sum of products}}{\text{Sum of squared differences for }x} $
$ = \frac{176}{11.3}$
$ \approx 15.6$
Now, let's calculate the $y$-intercept of the line.
$y$-intercept $ = \text{mean of } y - \text{slope }\times\text{mean of } x$
$ = 40 - 15.6 \times 10.6$
$ \approx -6.64$
Therefore, the equation of the line of best fit is:
$y = 15.6x - 6.6$
Rounded to the nearest tenth:
$y = 15.6x - 6.6$
ind an equation of the line of best fit for the data. Round the slope and the $y$ -intercept to the nearest tenth, if necessary.
x 7 8 10 13 15
y 25 29 41 48 57
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