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. Year 1995 1999 2000 2001 2002 2003 2004 Full Service 82 111 124 131 139 146 156 Fast Food 107 111 113 118 120 124 127 Question content area bottom Part 1 a) 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 a line that best fits the given data points. We can use the equation y = mx + b, where y represents the revenue in billions of dollars and x represents the number of years after 1990.
Using the given data points, we can calculate the values of m and b:
x = [0, 5, 9, 10, 11, 12, 13]
y = [82, 111, 124, 131, 139, 146, 156]
To find the values of m and b, we can use the formula:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
b = (Σy - mΣx) / n
where Σ denotes the sum of, n is the number of data points, Σxy is the sum of the product of x and y, ΣxΣy is the product of the sum of x and the sum of y, Σx^2 is the sum of the squares of x, and (Σx)^2 is the square of the sum of x.
Let's calculate the values:
n = 7
Σx = 0 + 5 + 9 + 10 + 11 + 12 + 13 = 60
Σy = 82 + 111 + 124 + 131 + 139 + 146 + 156 = 889
Σxy = (0*82) + (5*111) + (9*124) + (10*131) + (11*139) + (12*146) + (13*156) = 7434
Σx^2 = (0^2) + (5^2) + (9^2) + (10^2) + (11^2) + (12^2) + (13^2) = 515
Plugging these values into the formulas, we get:
m = (7*7434 - 60*889) / (7*515 - (60)^2) = 6.7 (approx.)
b = (889 - 6.7*60) / 7 = 75.3 (approx.)
Therefore, the linear regression equation for the "full service" restaurant is:
y = 6.7x + 75.3