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: 92, 116, 129, 140, 144, 152, 156
Fast food: 99, 107, 112, 118, 123, 127, 132
Write a linear regression equation for the "full service" restaurant.
To find the linear regression equation for the "full service" restaurant, we need to determine the equation of the line that best fits the data points. We can use the least squares regression method to achieve this.
First, let's calculate the mean of the x-values (years) and the mean of the y-values (revenue) to find the values around which the data points tend to cluster:
Mean of x-values (years):
(1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004) / 7 = 2001
Mean of y-values (revenue):
(92 + 116 + 129 + 140 + 144 + 152 + 156) / 7 = 134
Now, let's find the slope of the regression line (b) using the formula:
b = Σ((xi - x̄)(yi - ȳ)) / Σ(xi - x̄)^2
Where:
xi represents each individual x-value (year)
x̄ represents the mean of the x-values (years)
yi represents each individual y-value (revenue)
ȳ represents the mean of the y-values (revenue)
Σ denotes the sum of the values over the range.
Using the given data:
Σ((xi - x̄)(yi - ȳ)) = (1995 - 2001)(92 - 134) + (1999 - 2001)(116 - 134) + (2000 - 2001)(129 - 134) + (2001 - 2001)(140 - 134) + (2002 - 2001)(144 - 134) + (2003 - 2001)(152 - 134) + (2004 - 2001)(156 - 134)
= (-6)(-42) + (-2)(-18) + (-1)(-5) + (0)(6) + (1)(10) + (2)(18) + (3)(22)
= 257
Σ(xi - x̄)^2 = (1995 - 2001)^2 + (1999 - 2001)^2 + (2000 - 2001)^2 + (2001 - 2001)^2 + (2002 - 2001)^2 + (2003 - 2001)^2 + (2004 - 2001)^2
= (-6)^2 + (-2)^2 + (-1)^2 + (0)^2 + (1)^2 + (2)^2 + (3)^2
= 44
b = Σ((xi - x̄)(yi - ȳ)) / Σ(xi - x̄)^2 = 257 / 44 = 5.841
Now we can use the slope (b) to find the intercept (a) using the formula:
a = ȳ - b(x̄)
Where:
a represents the y-intercept of the regression line.
a = 134 - 5.841(2001) = -10,524.841
Thus, the linear regression equation for the "full service" restaurant is:
y = 5.841x - 10,524.841
Where y represents the annual revenue in billions of dollars and x represents the year (with x=0 representing 1990).