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 0 1 2 3 4
y 75 91 101 109 129
y =

Explanation:

Step1: Calculate the mean of x and y

First, find the mean of \( x \) values: \( \bar{x}=\frac{0 + 1+2 + 3+4}{5}=\frac{10}{5} = 2 \)
Then, find the mean of \( y \) values: \( \bar{y}=\frac{75+91 + 101+109+129}{5}=\frac{505}{5}=101 \)

Step2: Calculate the slope (m)

The formula for the slope of the line of best fit is \( m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}} \)

Calculate \( (x_i - \bar{x})(y_i - \bar{y}) \) for each \( i \):

• For \( x = 0,y = 75 \): \( (0 - 2)(75 - 101)=(-2)(-26) = 52 \)
• For \( x = 1,y = 91 \): \( (1 - 2)(91 - 101)=(-1)(-10)=10 \)
• For \( x = 2,y = 101 \): \( (2 - 2)(101 - 101)=0\times0 = 0 \)
• For \( x = 3,y = 109 \): \( (3 - 2)(109 - 101)=1\times8 = 8 \)
• For \( x = 4,y = 129 \): \( (4 - 2)(129 - 101)=2\times28 = 56 \)

Sum of these products: \( 52+10 + 0+8 + 56=126 \)

Calculate \( (x_i - \bar{x})^2 \) for each \( i \):

• For \( x = 0 \): \( (0 - 2)^2=4 \)
• For \( x = 1 \): \( (1 - 2)^2=1 \)
• For \( x = 2 \): \( (2 - 2)^2=0 \)
• For \( x = 3 \): \( (3 - 2)^2=1 \)
• For \( x = 4 \): \( (4 - 2)^2=4 \)

Sum of these squares: \( 4 + 1+0 + 1+4 = 10 \)

Then, \( m=\frac{126}{10}=12.6 \)

Step3: Calculate the y - intercept (b)

Using the formula \( \bar{y}=m\bar{x}+b \), we can solve for \( b \)
Substitute \( \bar{y}=101 \), \( m = 12.6 \) and \( \bar{x}=2 \) into the formula:
\( 101=12.6\times2 + b \)
\( 101 = 25.2+b \)
Subtract 25.2 from both sides: \( b=101 - 25.2 = 75.8 \)

Step4: Write the equation of the line of best fit

The equation of a line is \( y=mx + b \), substituting \( m = 12.6 \) and \( b = 75.8 \) we get \( y = 12.6x+75.8 \)

Answer:

\( y = 12.6x + 75.8 \)