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

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.
 Write a linear regression equation for the​ "full service" restaurant.
year Full Service
1995 95
1999 116
2000 131
2001 139
2002 144
2003 151
2004 156

GPT 3.5 · Answer

To find the linear regression equation for the "full service" restaurant, we need to find the equation of a line that best fits the given data points.

First, we can assign the years as the independent variable x and the revenue as the dependent variable y.

Let's create a table with the given data:

Year (x) | Revenue (y)
----------------------
1995 | 95
1999 | 116
2000 | 131
2001 | 139
2002 | 144
2003 | 151
2004 | 156

Next, we can use the least squares method to find the equation of the line:

1. Calculate the mean (average) of x and y:

mean(x) = (1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004) / 7 = 2001
mean(y) = (95 + 116 + 131 + 139 + 144 + 151 + 156) / 7 = 135.86

2. Calculate the deviations from the mean for each x and y:

(x - mean(x)) | (y - mean(y))
--------------------------------
-6 | -40.86
-2 | -19.86
-1 | -4.86
0 | 3.14
1 | 8.14
2 | 15.14
3 | 20.14

3. Calculate the sum of the products of the deviations:

Σ((x - mean(x))(y - mean(y))) = (-6 * -40.86) + (-2 * -19.86) + (-1 * -4.86) + (0 * 3.14) + (1 * 8.14) + (2 * 15.14) + (3 * 20.14) = 739.78

4. Calculate the sum of the squared deviations for x:

Σ((x - mean(x))^2) = (-6)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2 = 39

5. Use the formulas for the slope (b) and y-intercept (a) of the regression line:

b = Σ((x - mean(x))(y - mean(y))) / Σ((x - mean(x))^2)
= 739.78 / 39
≈ 18.94

a = mean(y) - b * mean(x)
= 135.86 - 18.94 * 2001
≈ -33,123.14

Therefore, the linear regression equation for the "full service" restaurant is:

y = -33,123.14 + 18.94x

Note: The units for x are years (e.g., x = 0 represents the year 1990), and the units for y are billions of dollars.