The annual profits for a company are given in the following table, where x represents the number of years since 1995, and y represents the profit in thousands of dollars. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, estimate the calendar year in which the profits would reach 244 thousand dollars.
Years since 1995 (x)	Profits (y)
(in thousands of dollars)
0	96
1	122
2	147
3	147

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.

Explore the linear regression equation derived from annual profit data to estimate future profits. Learn how to calculate slope, y-intercept, and predict profits for specific years. Suitable for high school students.

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