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The problem is asking for a linear regression model based on the cost per unit over eight years. To construct a linear regression model, we need to estimate the slope and intercept from the given data. Here's a brief step-by-step outline for finding the linear regression equation and the average cost increase per year:

### Step-by-Step Approach:
1. **Extract Data**: 
   - Year (independent variable, $x$): [1, 2, 3, 4, 5, 6, 7, 8]
   - Cost/Unit (dependent variable, $y$): [15.00, 18.00, 22.40, 20.50, 19.80, 23.00, 24.40, 30.40]

2. **Regression Formula**:
   The linear regression equation is typically given by:
   \[
   y = mx + b
   \]
   where:
   - $m$ is the slope (rate of change in cost per year, i.e., the average cost increase),
   - $b$ is the y-intercept (the initial cost when year = 0).

3. **Slope and Intercept Calculation**:
   The slope $m$ and intercept $b$ are calculated using the following formulas:
   \[
   m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
   \]
   \[
   b = \frac{\sum{y} - m\sum{x}}{n}
   \]
   where $n$ is the number of data points, $\sum{xy}$ is the sum of the products of $x$ and $y$, and $\sum{x^2}$ is the sum of squares of $x$.

Let me calculate the values of the slope, intercept, and the average cost increase per year.The linear regression equation for the given data is:

\[
\text{Unit Cost} = 1.6774 \times \text{Year} + 14.1393
\]

The average cost increase per year is approximately **$1.6774**.

Would you like further clarification or more details on this? Here are five related questions you can explore:

1. How do you calculate the correlation coefficient for this data?
2. Can we predict the cost per unit in year 10 using this model?
3. How does rounding affect the precision of the regression model?
4. What assumptions are we making with a linear regression model?
5. How would outliers impact the slope and intercept in this scenario?

**Tip:** When performing regression analysis, it's crucial to check the residuals to ensure the linear model fits well!

The president of a small manufacturing firm is concerned about the continual growth in manufacturing costs in the past several years. The data series of the cost per unit for the firm’s leading product over the past eight years is as follows: [Year, Cost/Unit: (1, $15.00), (2, $18.00), (3, $22.40), (4, $20.50), (5, $19.80), (6, $23.00), (7, $24.40), (8, $30.40)]. Use Excel to develop a simple linear regression model for these data. What average cost increase has the firm been realizing per year?

Learn how to use linear regression to model the cost per unit over eight years for a manufacturing firm. This analysis uses Excel to calculate the average yearly cost increase and provides a detailed step-by-step solution. Suitable for students in Grades 9-12 or college-level statistics courses.

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