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find regression equation and final answer

To solve this, we need to find the linear regression equation based on the data given. Then, we’ll use this equation to predict the profit for the year 2013.

### Step 1: Identify the Data Points
The data provided shows years \( x \) and profits \( y \):
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 161 \\
1 & 134 \\
2 & 197 \\
3 & 227 \\
4 & 209 \\
5 & 252 \\
\hline
\end{array}
\]

### Step 2: Calculate the Linear Regression Equation
The linear regression equation has the form:
\[
y = mx + b
\]
where:
- \( m \) is the slope, calculated as:
  \[
  m = \frac{n \sum(xy) - \sum x \sum y}{n \sum(x^2) - (\sum x)^2}
  \]
- \( b \) is the y-intercept, calculated as:
  \[
  b = \frac{\sum y - m \sum x}{n}
  \]
where \( n \) is the number of data points (in this case, \( n = 6 \)).

### Step 3: Substitute and Compute Values
Using these formulas and substituting the values from the table, we compute \( m \) and \( b \) to get the linear regression equation. I’ll go ahead with these calculations now.The linear regression equation based on the given data is:
\[
y = 20.3x + 146.0
\]

To find the profit for the year 2013, we substitute \( x = 10 \) (since 2013 is 10 years after 2003):
\[
y = 20.3(10) + 146.0 = 349 \text{ thousand dollars}
\]

### Final Answer:
- **Regression Equation**: \( y = 20.3x + 146.0 \)
- **Projected Profit for 2013**: 349 thousand dollars

Would you like further details on the calculation steps, or do you have any questions?

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Here are some related questions for practice:
1. How would the regression equation change if another year of data were added?
2. What does the slope of the regression equation represent in this context?
3. How could you use the regression equation to predict profits for years beyond 2013?
4. How would the equation change if profits dropped consistently instead?
5. Can you calculate the residuals for each data point to check the accuracy of the model?

**Tip:** When using a regression line to make predictions, ensure the data points have a consistent trend, as predictions are less reliable outside the observed range.

The annual profits for a company are given in the following table, where x represents the number of years since 2003, 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 tenth. Using this equation, find the projected profit (in thousands of dollars) for 2013, rounded to the nearest thousand dollars.

Learn how to calculate the linear regression equation for a dataset of annual profits from 2003 to 2008 and use it to project profits for 2013. This problem involves finding the slope and intercept, then applying the equation to predict future profits. Suitable for students in Grades 10-12.

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