Math Problem Statement
The annual profits for a company are given in the following table, where x represents the number of years since 2010, 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, find the projected profit (in thousands of dollars) for 2019, rounded to the nearest thousand dollars. Years since 2010 (x) Profits (y) (in thousands of dollars) 00 5555 11 114114 22 162162 33 180180 44 161161 55 239239 Answer
Regression Equation:
Final Answer: thousand dollars
Solution
To solve this, we need to find the linear regression equation of the form:
Where:
- is the profit (in thousands of dollars).
- is the number of years since 2010.
- is the slope of the line.
- is the y-intercept.
The table of data is given as:
\hline x & y \, (\text{in thousands}) \\ \hline 0 & 5555 \\ 1 & 114114 \\ 2 & 162162 \\ 3 & 180180 \\ 4 & 161161 \\ 5 & 239239 \\ \hline \end{array}$$ --- ### Step 1: Compute the slope ($$m$$) and intercept ($$b$$) The formulas for $$m$$ and $$b$$ are: $$m = \frac{n \sum(xy) - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ #### Compute necessary sums: Let’s compute the required quantities: - $$n = 6$$ (number of data points) - $$\sum x = 0 + 1 + 2 + 3 + 4 + 5 = 15$$ - $$\sum y = 5555 + 114114 + 162162 + 180180 + 161161 + 239239 = 862411$$ - $$\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 0 + 1 + 4 + 9 + 16 + 25 = 55$$ - $$\sum xy = (0 \cdot 5555) + (1 \cdot 114114) + (2 \cdot 162162) + (3 \cdot 180180) + (4 \cdot 161161) + (5 \cdot 239239) = 0 + 114114 + 324324 + 540540 + 644644 + 1196195 = 2828817$$ #### Plug values into $$m$$: $$m = \frac{n \sum(xy) - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$$ $$m = \frac{6(2828817) - (15)(862411)}{6(55) - 15^2}$$ $$m = \frac{16972902 - 12936165}{330 - 225}$$ $$m = \frac{4036737}{105} \approx 38445.12$$ #### Plug values into $$b$$: $$b = \frac{\sum y - m \sum x}{n}$$ $$b = \frac{862411 - 38445.12(15)}{6}$$ $$b = \frac{862411 - 576676.8}{6}$$ $$b = \frac{285734.2}{6} \approx 47622.37$$ --- ### Final Regression Equation: $$y = 38445.12x + 47622.37$$ --- ### Step 2: Project the profit for 2019 2019 corresponds to $$x = 9$$ (since $$2019 - 2010 = 9$$). Substitute $$x = 9$$ into the regression equation: $$y = 38445.12(9) + 47622.37$$ $$y \approx 346006.08 + 47622.37 = 393628.45$$ Rounded to the nearest thousand dollars: $$\boxed{394000 \, \text{thousand dollars}}$$ --- ### Final Answers: - **Regression Equation**: $$y = 38445.12x + 47622.37$$ - **Projected Profit for 2019**: $$394 \, \text{thousand dollars}$$ --- Let me know if you'd like further clarifications or more detailed steps! 😊 ### Related Questions: 1. How do you calculate the slope ($$m$$) for any linear regression manually? 2. What does the intercept ($$b$$) signify in a real-world context like this? 3. How would predictions differ if we used a non-linear regression model? 4. How can you use technology (e.g., calculators or software) to speed up regression calculations? 5. What are the limitations of using linear regression for this dataset? --- **Tip**: Always check if the data fits a linear trend before applying linear regression—use scatter plots to visualize!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Statistics
Algebra
Formulas
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
b = (Σy - mΣx) / n
y = mx + b
Theorems
Least Squares Method
Suitable Grade Level
Grades 10-12