To determine the linear regression equation, we first need to find the equation in the form:

$y = mx + b$

where:

• $m$ is the slope.
• $b$ is the y-intercept.

The data points given are:

• $(x_1, y_1) = (0, 96)$
• $(x_2, y_2) = (1, 122)$
• $(x_3, y_3) = (2, 147)$
• $(x_4, y_4) = (3, 147)$

Step 1: Calculate the Slope ($m$)

The slope $m$ is given by:

$m = \frac{n\sum{xy} - (\sum{x})(\sum{y})}{n\sum{x^2} - (\sum{x})^2}$

Where:

• $n$ is the number of data points.
• $\sum{xy}$ is the sum of the products of each $x$ and corresponding $y$.
• $\sum{x}$ is the sum of all $x$ values.
• $\sum{y}$ is the sum of all $y$ values.
• $\sum{x^2}$ is the sum of the squares of all $x$ values.

Step 2: Calculate the Intercept ($b$)

The intercept $b$ is given by:

$b = \frac{\sum{y} - m\sum{x}}{n}$

Calculations:

First, let's compute the sums:

• $\sum{x} = 0 + 1 + 2 + 3 = 6$
• $\sum{y} = 96 + 122 + 147 + 147 = 512$
• $\sum{xy} = (0)(96) + (1)(122) + (2)(147) + (3)(147) = 0 + 122 + 294 + 441 = 857$
• $\sum{x^2} = 0^2 + 1^2 + 2^2 + 3^2 = 0 + 1 + 4 + 9 = 14$

Now, using the slope formula:

$m = \frac{4(857) - (6)(512)}{4(14) - (6)^2} = \frac{3428 - 3072}{56 - 36} = \frac{356}{20} = 17.8$

Next, calculate the intercept $b$:

$b = \frac{512 - 17.8 \times 6}{4} = \frac{512 - 106.8}{4} = \frac{405.2}{4} = 101.3$

The Linear Regression Equation:

The linear regression equation is:

$y = 17.8x + 101.3$

Step 3: Estimate the Year in Which Profits Reach 244 Thousand Dollars

Set $y = 244$ and solve for $x$:

$244 = 17.8x + 101.3$

$17.8x = 244 - 101.3$

$17.8x = 142.7$

$x = \frac{142.7}{17.8} \approx 8.02$

So, the company is estimated to reach a profit of 244 thousand dollars approximately 8 years after 1995, which corresponds to the calendar year 2003.

Would you like more details on any part of the solution or have any questions? Here are some related problems:

1. How would the equation change if the profits were in millions instead of thousands?
2. What would the equation be if the profits had grown at a constant percentage instead of linearly?
3. How would the estimate change if the data points included an outlier?
4. Can the data be modeled more accurately with a quadratic equation?
5. How would the equation be affected if you started tracking the years from 2000 instead of 1995?

Tip: Linear regression assumes a straight-line relationship. If the actual trend isn’t linear, a different model might be more appropriate.