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 xequals
0
represent the year 1990.
Year
1995
1999
2000
2001
2002
2003
2004
Full Service
88
116
134
141
146
153
161
Fast Food
96
102
106
112
116
120
122
Question content area bottom
Part 1
a) Write a linear regression equation for the "full service" restaurant.
yequals
8.255 x plus 46.872
(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.)
Part 2
b) Now write a linear regression equation for the "fast food" restaurant.
yequals
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.)
In order to write a linear regression equation for the "fast food" restaurant, we need to find the equation of a line that best fits the given data points.
Using the given data:
Year (x): 1990, 1995, 1999, 2000, 2001, 2002, 2003, 2004
Fast Food Revenue (y): 96, 102, 106, 112, 116, 120, 122
We can calculate the equation using linear regression. Let's denote the slope as m and the y-intercept as b.
To find the slope (m):
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
Where:
- n is the number of data points (in this case, 8)
- Σxy is the sum of the products of x and y (1990*96 + 1995*102 + 1999*106 + 2000*112 + 2001*116 + 2002*120 + 2003*122 + 2004*?)
- Σx is the sum of all x values (1990 + 1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004)
- Σy is the sum of all y values (96 + 102 + 106 + 112 + 116 + 120 + 122 + ?)
- Σx^2 is the sum of the squares of all x values (1990^2 + 1995^2 + 1999^2 + 2000^2 + 2001^2 + 2002^2 + 2003^2 + 2004^2)
After calculating the respective values, we find:
nΣxy = (1990*96) + (1995*102) + (1999*106) + (2000*112) + (2001*116) + (2002*120) + (2003*122) + (2004*y)
= 15409342 + 20178990 + 21274994 + 22448000 + 23119016 + 24062440 + 25119606 + (2004*y)
Σx = 1990 + 1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004
= 15994
Σy = 96 + 102 + 106 + 112 + 116 + 120 + 122 + y
= 794 + y
Σx^2 = (1990^2) + (1995^2) + (1999^2) + (2000^2) + (2001^2) + (2002^2) + (2003^2) + (2004^2)
= 3960100 + 3980025 + 3992001 + 4000000 + 4004001 + 4008004 + 4012009 + 4016016
Substituting these values into the slope formula:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
= (15409342 + 20178990 + 21274994 + 22448000 + 23119016 + 24062440 + 25119606 + (2004*y) - (15994 * (794 + y))) / (8 * 4016016 - (15994)^2)
Simplifying:
m = (167312288 + 2004y - 12686636 - 12717136y) / (32128128 - 255872036)
= (-10854528 - 10797032y) / (-223743908)
= (5427264 + 5398516y) / 111871954
Therefore, the linear regression equation for the "fast food" restaurant is:
y = (5427264 + 5398516x) / 111871954