Question

find 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
y =

Explanation:

Step1: Calculate the mean of x and y

First, find the mean of \( x \) values: \( x = [7, 8, 10, 13, 15] \)
\( \bar{x} = \frac{7 + 8 + 10 + 13 + 15}{5} = \frac{53}{5} = 10.6 \)
Then, find the mean of \( y \) values: \( y = [25, 29, 41, 48, 57] \)
\( \bar{y} = \frac{25 + 29 + 41 + 48 + 57}{5} = \frac{200}{5} = 40 \)

Step2: Calculate the slope (m)

The formula for the slope of the line of best fit is:
\( m = \frac{\sum_{i = 1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i - \bar{x})^2} \)
First, calculate \( (x_i - \bar{x})(y_i - \bar{y}) \) for each \( i \):

• For \( x_1 = 7, y_1 = 25 \): \( (7 - 10.6)(25 - 40) = (-3.6)(-15) = 54 \)
• For \( x_2 = 8, y_2 = 29 \): \( (8 - 10.6)(29 - 40) = (-2.6)(-11) = 28.6 \)
• For \( x_3 = 10, y_3 = 41 \): \( (10 - 10.6)(41 - 40) = (-0.6)(1) = -0.6 \)
• For \( x_4 = 13, y_4 = 48 \): \( (13 - 10.6)(48 - 40) = (2.4)(8) = 19.2 \)
• For \( x_5 = 15, y_5 = 57 \): \( (15 - 10.6)(57 - 40) = (4.4)(17) = 74.8 \)

Sum these values: \( 54 + 28.6 - 0.6 + 19.2 + 74.8 = 176 \)
Next, calculate \( (x_i - \bar{x})^2 \) for each \( i \):

• For \( x_1 = 7 \): \( (7 - 10.6)^2 = (-3.6)^2 = 12.96 \)
• For \( x_2 = 8 \): \( (8 - 10.6)^2 = (-2.6)^2 = 6.76 \)
• For \( x_3 = 10 \): \( (10 - 10.6)^2 = (-0.6)^2 = 0.36 \)
• For \( x_4 = 13 \): \( (13 - 10.6)^2 = (2.4)^2 = 5.76 \)
• For \( x_5 = 15 \): \( (15 - 10.6)^2 = (4.4)^2 = 19.36 \)

Sum these values: \( 12.96 + 6.76 + 0.36 + 5.76 + 19.36 = 45.2 \)
Now, calculate the slope: \( m = \frac{176}{45.2} \approx 3.9 \) (rounded to the nearest tenth)

Step3: Calculate the y-intercept (b)

Use the formula \( \bar{y} = m\bar{x} + b \)
Substitute \( \bar{y} = 40 \), \( m \approx 3.9 \), and \( \bar{x} = 10.6 \)
\( 40 = 3.9 \times 10.6 + b \)
\( 40 = 41.34 + b \)
\( b = 40 - 41.34 = -1.34 \approx -1.3 \) (rounded to the nearest tenth)

Step4: Write the equation of the line of best fit

The equation of a line is \( y = mx + b \), so substituting \( m \approx 3.9 \) and \( b \approx -1.3 \)
\( y = 3.9x - 1.3 \)

Answer:

\( y = 3.9x - 1.3 \)