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 represents the year 1990
a) Write a linear regression equation for the "full service" restaurant.
y=
Without the data provided, we cannot write the linear regression equation for the "full service" restaurant.
Year
1995
1999
2000
2001
2002
2003
2004
Full Service
96
124
135
142
147
151
159
Fast Food
105
109
113
116
119
123
127
To write a linear regression equation for the "full service" restaurant, we need to find the equation that best fits the relationship between the year (x) and the revenue (y) for the "full service" restaurant.
Using the given data, we can create a scatter plot and try to find a line that closely fits the data points. After finding the line, we can determine its equation.
The scatter plot of the "full service" restaurant data:
Year (x) Revenue (y)
1995 96
1999 124
2000 135
2001 142
2002 147
2003 151
2004 159
Based on the scatter plot, we can see that the points form a positive linear relationship. Let's find the equation of the line using regression analysis:
Step 1: Calculate the mean of x and y:
mean(x) = (1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004) / 7
mean(x) = 2001
mean(y) = (96 + 124 + 135 + 142 + 147 + 151 + 159) / 7
mean(y) = 138.71 (approx)
Step 2: Find the deviations (d) from the means for x and y:
d(x) = (x - mean(x))
d(y) = (y - mean(y))
d(x): -6, -2, -1, 0, 1, 2, 3
d(y): -42.71, -14.71, -3.71, 3.29, 8.29, 12.29, 20.29
Step 3: Calculate the sum of the product of deviations (d) from the means:
Σd(x) * d(y) = (-6 * -42.71) + (-2 * -14.71) + (-1 * -3.71) + (0 * 3.29) + (1 * 8.29) + (2 * 12.29) + (3 * 20.29)
Σd(x) * d(y) = 239.21
Step 4: Calculate the sum of squared deviations:
Σd(x)^2 = (-6)^2 + (-2)^2 + (-1)^2 + (0)^2 + (1)^2 + (2)^2 + (3)^2
Σd(x)^2 = 55
Step 5: Calculate the slope (b):
b = Σd(x) * d(y) / Σd(x)^2
b = 239.21 / 55
b = 4.35 (approx)
Step 6: Calculate the y-intercept (a):
a = mean(y) - b * mean(x)
a = 138.71 - 4.35 * 2001
a = -8626.71 (approx)
The linear regression equation for the "full service" restaurant can be written as:
y = 4.35x - 8626.71
Therefore, the linear regression equation for the "full service" restaurant is:
y = 4.35x - 8626.71