The annual profits for a company are given in the following table, where x represents the number of years since 2002, 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 2010, rounded *to the nearest thousand dollars*.

Years since 2002 (x)Profits (y)
(in thousands of dollars)

0 135

1 160

2 157

3 192

4 192

5 227

### Step 1: Set up the data
The given table provides:

\[
\begin{array}{|c|c|}
\hline
x & y \, (\text{Profit in thousands}) \\
\hline
0 & 135 \\
1 & 160 \\
2 & 157 \\
3 & 192 \\
4 & 192 \\
5 & 227 \\
\hline
\end{array}
\]

Where $ x $ is the number of years since 2002, and $ y $ is the profit in thousands of dollars.

### Step 2: Use the formula for linear regression
The equation of a straight line (linear regression equation) is:
\[
y = mx + b
\]
Where:
- $ m $ is the slope of the line, given by:
\[
m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]
- $ b $ is the y-intercept, given by:
\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]
Where $ n $ is the number of data points.

### Step 3: Calculate the sums needed for the formulas
Let me calculate all the necessary values (like sums of $ x $, $ y $, $ xy $, etc.) and then compute the linear regression equation.The linear regression equation that best fits the data is:

\[
y = 16.89x + 134.95
\]
(rounded to the nearest hundredth).

### Step 4: Use the equation to project the profit for 2010
Since $ x $ represents the number of years since 2002, the year 2010 corresponds to $ x = 2010 - 2002 = 8 $.

Let’s calculate the projected profit for $ x = 8 $.The projected profit for the year 2010, based on the linear regression model, is approximately **$270,000** (rounded to the nearest thousand dollars).

Would you like more details on this or have any questions?

Here are 5 related questions to explore:
1. How do you interpret the slope of the linear regression in terms of company growth?
2. What would the projected profit be for the year 2015?
3. Can you calculate the residuals for this data and check the model's accuracy?
4. How would the linear regression change if you included more years of data?
5. What happens to the regression model if one of the data points significantly changes?

**Tip:** Always check the goodness of fit (e.g., using $R^2$) when working with linear regression models to measure how well the model explains the data.

The annual profits for a company are given in the following table, where x represents the number of years since 2002, 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 2010, rounded to the nearest thousand dollars.

Years since 2002 (x) Profits (y)
(in thousands of dollars)
0 135
1 160
2 157
3 192
4 192
5 227

Calculate the linear regression equation for annual company profits (in thousands of dollars) from 2002 to 2007, and find the projected profit for 2010. Learn how to use the linear regression formula, rounding coefficients to the nearest hundredth, with a step-by-step solution for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation