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.
" Full Service"
87
117
125
131
136
139
146
Write a linear regression equation for the "full service" restaurant.
To write a linear regression equation for the "full service" restaurant, we need to find the equation of the line that best fits the data points provided.
First, let's assign the years as the independent variable (x) and the revenue in billions of dollars as the dependent variable (y).
Based on the given data points, we have the following values:
x = 0, 1, 2, 3, 4, 5, 6
y = 87, 117, 125, 131, 136, 139, 146
To find the equation, we need to calculate the slope (m) and the y-intercept (b).
Step 1: Calculate the slope (m):
m = (Σxy - n(Σx)(Σy)) / (Σx^2 - n(Σx)^2)
Σxy = (0*87) + (1*117) + (2*125) + (3*131) + (4*136) + (5*139) + (6*146)
= 1467
Σx = 0 + 1 + 2 + 3 + 4 + 5 + 6
= 21
Σy = 87 + 117 + 125 + 131 + 136 + 139 + 146
= 881
Σx^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) + (6^2)
= 91
n = number of data points = 7
m = (1467 - 7(21)(881)) / (91 - 7(21)^2)
= (1467 - 9873) / (91 - 1029)
= -8406 / -938
≈ 8.96
Step 2: Calculate the y-intercept (b):
b = (Σy - m(Σx)) / n
b = (881 - 8.96(21)) / 7
= (881 - 189.76) / 7
= 691.24 / 7
≈ 98.75
Step 3: Write the linear regression equation:
y = mx + b
y = 8.96x + 98.75
Therefore, the linear regression equation for the "full service" restaurant is y = 8.96x + 98.75.