Step 1: Organize the Data

We are tasked with finding the linear regression equation $y = mx + b$, where:

• $x$ is the number of years since 2014,
• $y$ is the profit in thousands of dollars.

Data Table

$x$	$y$
0	85
1	86
2	96
3	152
4	172

---

Step 2: Compute the Necessary Summations

To calculate the linear regression coefficients, we need:
$m = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}$ $b = \frac{\sum y - m\sum x}{n}$

Intermediate values:

• $\sum x = 0 + 1 + 2 + 3 + 4 = 10$,
• $\sum y = 85 + 86 + 96 + 152 + 172 = 591$,
• $\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 30$,
• $\sum xy = (0)(85) + (1)(86) + (2)(96) + (3)(152) + (4)(172) = 1118$,
• $n = 5$.

---

Step 3: Solve for $m$ (Slope)

Substitute values into the slope formula: $m = \frac{5(1118) - (10)(591)}{5(30) - (10)^2}$ $m = \frac{5590 - 5910}{150 - 100}$ $m = \frac{-320}{50} = -6.4$

---

Step 4: Solve for $b$ (Y-Intercept)

Substitute into the intercept formula: $b = \frac{\sum y - m\sum x}{n}$ $b = \frac{591 - (-6.4)(10)}{5}$ $b = \frac{591 + 64}{5} = \frac{655}{5} = 131$

---

Step 5: Write the Linear Regression Equation

The linear regression equation is: $y = -6.4x + 131$

---

Step 6: Project Profit for 2025

In 2025, $x = 2025 - 2014 = 11$. Substitute $x = 11$ into the equation: $y = -6.4(11) + 131$ $y = -70.4 + 131 = 60.6 \text{ (in thousands of dollars)}.$ Rounding to the nearest thousand dollars: $y \approx 61 \, \text{thousand dollars}.$

---

Final Answer:

• Regression Equation: $y = -6.4x + 131$
• Projected Profit for 2025: $\mathbf{61 \, \text{thousand dollars}}$

---

Would you like detailed steps on any of these calculations?
Here are five related questions for exploration:

1. How is the linear regression formula derived?
2. What is the significance of the slope $m$ in real-world terms?
3. How do we interpret the y-intercept in this context?
4. What happens to the accuracy of the model if more data points are added?
5. How would a quadratic regression compare to this linear model for profit prediction?

Tip: Always plot the data points and the regression line to visually assess how well the model fits!