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
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Part 1
a) Write a linear regression equation for the "full service" 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.)
The linear regression equation for the "full service" restaurant is y = 3.067x + 88.
no
My apologies for the incorrect response. Let me calculate the correct linear regression equation for the "full service" restaurant.
To find the linear regression equation, we need to calculate the slope and y-intercept. Let's use the given data points:
Year:
1990, 1995, 1999, 2000, 2001, 2002, 2003, 2004
Revenue in billions of dollars:
88, 116, 134, 141, 146, 153, 161
We can use these values to calculate the slope:
slope (m) = (Σ(x*y) - n*(mean(x)*mean(y))) / (Σ(x^2) - n*(mean(x)^2))
x-values: 0, 5, 9, 10, 11, 12, 13, 14
y-values: 88, 116, 134, 141, 146, 153, 161
Calculating the sum of these values:
Σ(x) = 0 + 5 + 9 + 10 + 11 + 12 + 13 + 14 = 74
Σ(y) = 88 + 116 + 134 + 141 + 146 + 153 + 161 = 939
Σ(x*y) = (0*88) + (5*116) + (9*134) + (10*141) + (11*146) + (12*153) + (13*161) = 12043
Σ(x^2) = (0^2) + (5^2) + (9^2) + (10^2) + (11^2) + (12^2) + (13^2) + (14^2) = 634
Calculating the mean:
mean(x) = Σ(x) / n = 74 / 8 = 9.25
mean(y) = Σ(y) / n = 939 / 8 = 117.375
Substituting the values into the formula:
slope (m) = (12043 - 8*(9.25)*(117.375)) / (634 - 8*(9.25)^2) = 3.173
Now let's calculate the y-intercept:
y-intercept (b) = mean(y) - slope * mean(x) = 117.375 - 3.173 * 9.25 = 86.445
Therefore, the linear regression equation for the "full service" restaurant is:
y = 3.173x + 86.445 (rounded to three decimal places)