Question

The annual profits for a company are given in the following table, where x represents the number of years since 2009, 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 2021, rounded to the nearest thousand dollars.
Years since 2009 (x) Profits (y)
(in thousands of dollars)
00 4747
11 6464
22 6666
33 8989
44 9898
Answer
Attempt 1 out of 2

Regression Equation:

Final Answer:
thousand dollars

Answer 1

To find the linear regression equation from the given data, we first identify our data points:

- For \( x = 0 \), \( y = 4747 \)
- For \( x = 1 \), \( y = 6464 \)
- For \( x = 2 \), \( y = 6666 \)
- For \( x = 3 \), \( y = 8989 \)
- For \( x = 4 \), \( y = 9898 \)

We will perform linear regression using the least squares method to find the coefficients of the regression line in the form of \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

### Step 1: Calculate the necessary sums for regression

- \( n = 5 \) (number of data points)
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 4747 + 6464 + 6666 + 8989 + 9898 = 38764 \)
- \( \sum xy = 0 \cdot 4747 + 1 \cdot 6464 + 2 \cdot 6666 + 3 \cdot 8989 + 4 \cdot 9898 = 0 + 6464 + 13332 + 26967 + 39592 = 184255 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)

### Step 2: Calculate the slope \( m \) and y-intercept \( b \)

The formulas for the slope \( m \) and y-intercept \( b \) of the linear regression line are:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Substituting the calculated sums into the slope formula:

\[
m = \frac{5(184255) - (10)(38764)}{5(30) - (10)^2}
\]

Calculating \( m \):

\[
m = \frac{921275 - 387640}{150 - 100} = \frac{533635}{50} = 10672.7
\]

Next, we calculate the y-intercept \( b \):

\[
b = \frac{38764 - 10672.7(10)}{5} = \frac{38764 - 106727}{5} = \frac{-67963}{5} = -13592.6
\]

### Step 3: Write the regression equation

So, the linear regression equation is:

\[
y = 10672.7x - 13592.6
\]

### Step 4: Round coefficients

Rounding to the nearest hundredth:

\[
y = 10672.70x - 13592.60
\]

### Step 5: Projected profit for 2021

In 2021, \( x = 2021 - 2009 = 12 \).

Substituting \( x = 12 \) into the regression equation:

\[
y = 10672.7(12) - 13592.6
\]
\[
y = 128072.4 - 13592.6 = 113479.8
\]

### Final answer:

Rounding to the nearest thousand dollars, the projected profit for 2021 is:

\[
\text{Projected profit} = 113480 \text{ thousand dollars}
\]

**Regression Equation:** \( y = 10672.70x - 13592.60 \)

**Final Answer:** 113480 thousand dollars