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 99 127 145 152 156 163 171
Fast Food 110 118 125 131 136 139 142
Question content area bottom Part 1 a) Write a linear regression equation for the "full service" restaurant. y=enter your response here (Type an expression using x as the variable. Use integers or decimals for any numbers in the expression. Round to the nearest thousandth as needed.)
To find the linear regression equation for the "full service" restaurant, we can use the least squares method to find the equation of a line that best fits the data.
Let's use the formula for a linear regression equation: y = mx + b
To determine the values of m and b, we need to find the slope (m) and y-intercept (b).
We will use the given data points (x, y) = (0, 99), (5, 127), (9, 145), (10, 152), (11, 156), (12, 163), (13, 171).
First, let's find the slope (m):
m = (n∑xy - (∑x)(∑y))/(n∑x^2 - (∑x)^2)
n = 7 (number of data points)
∑x = 0 + 5 + 9 + 10 + 11 + 12 + 13 = 60
∑y = 99 + 127 + 145 + 152 + 156 + 163 + 171 = 1013
∑xy = (0*99) + (5*127) + (9*145) + (10*152) + (11*156) + (12*163) + (13*171) = 14926
∑x^2 = (0^2) + (5^2) + (9^2) + (10^2) + (11^2) + (12^2) + (13^2) = 455
m = (7*14926 - (60)(1013))/(7*455 - (60)^2)
m ≈ 196.933
Next, let's find the y-intercept (b):
b = (∑y - m(∑x))/n
b = (1013 - (196.933)(60))/7
b ≈ -51.081
Therefore, the linear regression equation for the "full service" restaurant is:
y = 196.933x - 51.081 (rounded to the nearest thousandth)