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To solve this problem, we need to find the linear regression equation of the form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the profit in thousands of dollars,
- [tex]\( x \)[/tex] is the number of years since 2012,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept of the line.
Given data points:
[tex]\[ (x_i, y_i): \quad (0, 127), (1, 126), (2, 151), (3, 169) \][/tex]
### Step 1: Calculate the Means
First, we calculate the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
[tex]\[ \bar{x} = \frac{\sum x_i}{n} = \frac{0 + 1 + 2 + 3}{4} = \frac{6}{4} = 1.5 \][/tex]
[tex]\[ \bar{y} = \frac{\sum y_i}{n} = \frac{127 + 126 + 151 + 169}{4} = \frac{573}{4} = 143.25 \][/tex]
### Step 2: Calculate the Slope (m) and Intercept (b)
The formulas for the slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] are:
[tex]\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
[tex]\[ b = \bar{y} - m\bar{x} \][/tex]
To find [tex]\( m \)[/tex], we first need to compute the components:
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (0-1.5)(127-143.25) + (1-1.5)(126-143.25) + (2-1.5)(151-143.25) + (3-1.5)(169-143.25) \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (-1.5)(-16.25) + (-0.5)(-17.25) + (0.5)(7.75) + (1.5)(25.75) \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 24.375 + 8.625 + 3.875 + 38.625 = 75.5 \][/tex]
Next, we find [tex]\( \sum (x_i - \bar{x})^2 \)[/tex]:
[tex]\[ \sum (x_i - \bar{x})^2 = (0-1.5)^2 + (1-1.5)^2 + (2-1.5)^2 + (3-1.5)^2 \][/tex]
[tex]\[ \sum (x_i - \bar{x})^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5 \][/tex]
Now, calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{75.5}{5} = 15.1 \][/tex]
Finally, calculate the intercept [tex]\( b \)[/tex]:
[tex]\[ b = \bar{y} - m\bar{x} = 143.25 - (15.1 \cdot 1.5) = 143.25 - 22.65 = 120.6 \][/tex]
### Step 3: Write the Linear Regression Equation
The linear regression equation is:
[tex]\[ y = 15.1x + 120.6 \][/tex]
### Step 4: Project the Profit for 2021
To find the profit for 2021, note that 2021 is 9 years since 2012 ([tex]\( x = 9 \)[/tex]):
[tex]\[ y = 15.1 \cdot 9 + 120.6 \][/tex]
[tex]\[ y = 135.9 + 120.6 = 256.5 \][/tex]
Rounding to the nearest thousand dollars:
[tex]\[ y \approx 257 \ \text{thousand dollars} \][/tex]
### Final Answer
- Regression Equation: [tex]\( y = 15.1x + 120.6 \)[/tex]
- Projected Profit for 2021: [tex]\( 257 \ \text{thousand dollars} \)[/tex]
Please fill in the answer slots as required:
- Regression Equation: [tex]\( y = 15.1x + 120.6 \)[/tex]
- Final Answer: [tex]\( 257 \ \text{thousand dollars} \)[/tex]
- [tex]\( y \)[/tex] is the profit in thousands of dollars,
- [tex]\( x \)[/tex] is the number of years since 2012,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept of the line.
Given data points:
[tex]\[ (x_i, y_i): \quad (0, 127), (1, 126), (2, 151), (3, 169) \][/tex]
### Step 1: Calculate the Means
First, we calculate the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
[tex]\[ \bar{x} = \frac{\sum x_i}{n} = \frac{0 + 1 + 2 + 3}{4} = \frac{6}{4} = 1.5 \][/tex]
[tex]\[ \bar{y} = \frac{\sum y_i}{n} = \frac{127 + 126 + 151 + 169}{4} = \frac{573}{4} = 143.25 \][/tex]
### Step 2: Calculate the Slope (m) and Intercept (b)
The formulas for the slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] are:
[tex]\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
[tex]\[ b = \bar{y} - m\bar{x} \][/tex]
To find [tex]\( m \)[/tex], we first need to compute the components:
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (0-1.5)(127-143.25) + (1-1.5)(126-143.25) + (2-1.5)(151-143.25) + (3-1.5)(169-143.25) \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (-1.5)(-16.25) + (-0.5)(-17.25) + (0.5)(7.75) + (1.5)(25.75) \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 24.375 + 8.625 + 3.875 + 38.625 = 75.5 \][/tex]
Next, we find [tex]\( \sum (x_i - \bar{x})^2 \)[/tex]:
[tex]\[ \sum (x_i - \bar{x})^2 = (0-1.5)^2 + (1-1.5)^2 + (2-1.5)^2 + (3-1.5)^2 \][/tex]
[tex]\[ \sum (x_i - \bar{x})^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5 \][/tex]
Now, calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{75.5}{5} = 15.1 \][/tex]
Finally, calculate the intercept [tex]\( b \)[/tex]:
[tex]\[ b = \bar{y} - m\bar{x} = 143.25 - (15.1 \cdot 1.5) = 143.25 - 22.65 = 120.6 \][/tex]
### Step 3: Write the Linear Regression Equation
The linear regression equation is:
[tex]\[ y = 15.1x + 120.6 \][/tex]
### Step 4: Project the Profit for 2021
To find the profit for 2021, note that 2021 is 9 years since 2012 ([tex]\( x = 9 \)[/tex]):
[tex]\[ y = 15.1 \cdot 9 + 120.6 \][/tex]
[tex]\[ y = 135.9 + 120.6 = 256.5 \][/tex]
Rounding to the nearest thousand dollars:
[tex]\[ y \approx 257 \ \text{thousand dollars} \][/tex]
### Final Answer
- Regression Equation: [tex]\( y = 15.1x + 120.6 \)[/tex]
- Projected Profit for 2021: [tex]\( 257 \ \text{thousand dollars} \)[/tex]
Please fill in the answer slots as required:
- Regression Equation: [tex]\( y = 15.1x + 120.6 \)[/tex]
- Final Answer: [tex]\( 257 \ \text{thousand dollars} \)[/tex]
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