To find the linear regression equation from the given data, we first identify our data points:
- For \( x = 0 \), \( y = 4747 \)
- For \( x = 1 \), \( y = 6464 \)
- For \( x = 2 \), \( y = 6666 \)
- For \( x = 3 \), \( y = 8989 \)
- For \( x = 4 \), \( y = 9898 \)
We will perform linear regression using the least squares method to find the coefficients of the regression line in the form of \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
Step 1: Calculate the necessary sums for regression
- \( n = 5 \) (number of data points)
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 4747 + 6464 + 6666 + 8989 + 9898 = 38764 \)
- \( \sum xy = 0 \cdot 4747 + 1 \cdot 6464 + 2 \cdot 6666 + 3 \cdot 8989 + 4 \cdot 9898 = 0 + 6464 + 13332 + 26967 + 39592 = 184255 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)
Step 2: Calculate the slope \( m \) and y-intercept \( b \)
The formulas for the slope \( m \) and y-intercept \( b \) of the linear regression line are:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{\sum y - m(\sum x)}{n} \]
Substituting the calculated sums into the slope formula:
\[ m = \frac{5(184255) - (10)(38764)}{5(30) - (10)^2} \]
Calculating \( m \):
\[ m = \frac{921275 - 387640}{150 - 100} = \frac{533635}{50} = 10672.7 \]
Next, we calculate the y-intercept \( b \):
\[ b = \frac{38764 - 10672.7(10)}{5} = \frac{38764 - 106727}{5} = \frac{-67963}{5} = -13592.6 \]
Step 3: Write the regression equation
So, the linear regression equation is:
\[ y = 10672.7x - 13592.6 \]
Step 4: Round coefficients
Rounding to the nearest hundredth:
\[ y = 10672.70x - 13592.60 \]
Step 5: Projected profit for 2021
In 2021, \( x = 2021 - 2009 = 12 \).
Substituting \( x = 12 \) into the regression equation:
\[ y = 10672.7(12) - 13592.6 \] \[ y = 128072.4 - 13592.6 = 113479.8 \]
Final answer:
Rounding to the nearest thousand dollars, the projected profit for 2021 is:
\[ \text{Projected profit} = 113480 \text{ thousand dollars} \]
Regression Equation: \( y = 10672.70x - 13592.60 \)
Final Answer: 113480 thousand dollars