Shown below are rental and leasing revenue figures for office machinery and equipment in the United States over a sever-year period according to the U.S. Census Bureau. Use these data to run a linear regression and then forecast the rental and leasing revenue for the year 2012.

2004 5,860
2005 6,632
2006 7,125
2007 6,000
2008 4,380
2009 3,326
2010 2,642

Just go to your favorite regression calculator, and if y(x) is the line estimating x years after 2004,

y = 7174.75 - 678.964x

So, for 2012,
y(8) = -972.77

It's not likely that we will get a negative value, and indeed we see that since the data rise and then fall, a linear model isn't very good in this case.

Y=71745-678.964x =6495.786

So for 2012,
y (8) = -972.77

To forecast the rental and leasing revenue for the year 2012 using linear regression, we can follow these steps:

Step 1: Assign the given data to variable X and Y:
X represents the years (2004, 2005, 2006, 2007, 2008, 2009, 2010)
Y represents the rental and leasing revenue (5,860, 6,632, 7,125, 6,000, 4,380, 3,326, 2,642)

Step 2: Plot the data points on a scatter plot to visualize the relationship between the years and the rental and leasing revenue. This will help us understand whether a linear regression model is appropriate for this data.

Step 3: Calculate the slope and intercept of the linear regression line using the least squares method. This will give us the equation of the line: Y = m*X + b

Step 4: Substitute the year 2012 into the equation calculated in step 3 to forecast the rental and leasing revenue for that year.

Let's walk through each step in detail:

Step 1: Assign the given data to variables X and Y:
X = [2004, 2005, 2006, 2007, 2008, 2009, 2010]
Y = [5,860, 6,632, 7,125, 6,000, 4,380, 3,326, 2,642]

Step 2: Plot the data points on a scatter plot:
Here is a graphical representation of the data points (X, Y):

2004 5,860
2005 6,632
2006 7,125 *
2007 6,000 *
2008 4,380 *
2009 3,326 *
2010 2,642

We can see that there is a general downward trend in the rental and leasing revenue over time, indicating that a linear regression model may be appropriate.

Step 3: Calculate the slope and intercept of the linear regression line:
To calculate the slope (m) and intercept (b) of the line, we will use the least squares method. The formula for calculating the slope (m) is:

m = (n * Σ(X*Y) - Σ(X) * Σ(Y)) / (n * Σ(X^2) - (Σ(X))^2)

where Σ represents the sum, and n is the number of data points.

Calculating the sum and squares:

Σ(X) = 14,057
Σ(Y) = 36,265
Σ(X*Y) = 199,803
Σ(X^2) = 20,058

Using the formula, we get:

m = (7 * 199,803 - 14,057 * 36,265) / (7 * 20,058 - (14,057)^2)
m ≈ -1105.714

The formula for calculating the intercept (b) is:

b = (Σ(Y) - m * Σ(X)) / n

Substituting the values, we get:

b = (36,265 - (-1105.714) * 14,057) / 7
b ≈ 2,304.286

So the equation of the linear regression line is:

Y = -1105.714X + 2,304.286

Step 4: Forecast the rental and leasing revenue for the year 2012:
To forecast the rental and leasing revenue for the year 2012, we substitute X = 2012 into the equation:

Y = -1105.714 * 2012 + 2,304.286
Y ≈ -2,233,714.768

The forecasted rental and leasing revenue for the year 2012 is approximately $2,233,714.768.

Note: Linear regression assumes a linear relationship between the independent variable (in this case, the year) and the dependent variable (the rental and leasing revenue). Keep in mind that there may be other factors not accounted for in this analysis that can affect the revenue.