In the United​ States, the revenue​ (money taken from​ sales) at a​ "full service" restaurant is increasing at a faster rate than the revenue at a​ "fast food" restaurant. The data below represent the annual revenue in billions of dollars for each type of restaurant. Use the data to answer the three questions below. Let x=0 represent the year 1990.

Question

year Full Service
1995 87 117 125 131 136 139 146 Write a linear regression equation for the​ "full service" restaurant.
1999
2000
2001
2002
2003
2004

Answer 1

To write a linear regression equation for the "full service" restaurant, we need to calculate the slope and y-intercept using the given data.

First, let's assign the year values as independent variable x and the revenue values as dependent variable y. We will use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

The given data for the "full service" restaurant is as follows:

Year (x): 1995, 1999, 2000, 2001, 2002, 2003, 2004
Revenue (y): 87, 117, 125, 131, 136, 139, 146

Let's pick two points from the data to calculate the slope:

Point 1: (x1, y1) = (1995, 87)
Point 2: (x2, y2) = (2004, 146)

The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)

Substituting the values, we get:
m = (146 - 87) / (2004 - 1995)
m = 59 / 9
m = 6.5556 (rounded to four decimal places)

Next, we need to calculate the y-intercept (b) using the formula:
b = y - mx

Using either of the two points, let's substitute the values:
87 = 6.5556(1995) + b

Simplifying the equation:
b = 87 - 6.5556(1995)
b = 87 - 13071.8916
b = -12984.8916 (rounded to four decimal places)

Therefore, the linear regression equation for the "full service" restaurant is:
y = 6.5556x - 12984.8916

Note: The years other than 1995, 1999, 2000, 2001, 2002, 2003, and 2004 are not provided, so they are not included in the equation.