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 write a linear regression equation for the "full service" restaurant, we need to find the equation of a straight line that best fits the data points for the revenue over time.
Given that x = 0 represents the year 1990, we will use x = 5 for the year 1995, and so on. Let's denote the year as x and the revenue as y.
The given data points are:
(5, 92), (9, 116), (10, 129), (11, 140), (12, 144), (13, 152), (14, 156).
Using these data points, we can use the least squares method to find the equation of the line:
Step 1: Calculate the means (average) of x and y:
mean(x) = (5 + 9 + 10 + 11 + 12 + 13 + 14) / 7 = 11
mean(y) = (92 + 116 + 129 + 140 + 144 + 152 + 156) / 7 = 133
Step 2: Calculate the sum of products (x - mean(x))(y - mean(y)) for each data point:
(5 - 11)(92 - 133) = -36
(9 - 11)(116 - 133) = -34
(10 - 11)(129 - 133) = -8
(11 - 11)(140 - 133) = 7
(12 - 11)(144 - 133) = 11
(13 - 11)(152 - 133) = 38
(14 - 11)(156 - 133) = 69
Step 3: Calculate the sum of squares (x - mean(x))^2 for each data point:
(5 - 11)^2 = 36
(9 - 11)^2 = 4
(10 - 11)^2 = 1
(11 - 11)^2 = 0
(12 - 11)^2 = 1
(13 - 11)^2 = 4
(14 - 11)^2 = 9
Step 4: Calculate the slope (b) of the line:
b = sum of products / sum of squares
b = (-36 - 34 - 8 + 7 + 11 + 38 + 69) / (36 + 4 + 1 + 0 + 1 + 4 + 9)
b = 135 / 55
b ≈ 2.45
Step 5: Calculate the y-intercept (a) of the line:
a = mean(y) - b * mean(x)
a = 133 - 2.45 * 11
a = 133 - 26.95
a ≈ 106.05
Therefore, the linear regression equation for the "full service" restaurant is:
y = 2.45x + 106.05
it says that this is incorrect. is there any other way to solve this?
Apologies for the incorrect response earlier. Let's try solving it again using a different approach.
We will use the method of least squares to find the equation of the line that best fits the data points for the "full service" restaurant.
First, let's assign the year as x and the revenue as y.
The data points are as follows:
(1995, 92), (1999, 116), (2000, 129), (2001, 140), (2002, 144), (2003, 152), (2004, 156)
To use linear regression, we need to standardize the years by subtracting the value of the first year (1990) from each year (x).
The standardized data points are:
(x-1995, y-92), (x-1990, y-116), (x-1990, y-129), (x-1990, y-140), (x-1990, y-144), (x-1990, y-152), (x-1990, y-156)
Let's denote the standardized years as X and the standardized revenue as Y.
The standardized data points are as follows:
(X, Y) = (-5, 0), (-1, 24), (0, 37), (1, 48), (2, 52), (3, 60), (4, 64)
Using these data points, we can calculate the slope and y-intercept of the line using the formulas:
slope (b) = (n*sum(X * Y) - sum(X) * sum(Y)) / (n*sum(X^2) - (sum(X))^2)
y-intercept (a) = mean(Y) - b * mean(X)
where n is the number of data points.
Calculating the necessary sums:
sum(X) = -5 + (-1) + 0 + 1 + 2 + 3 + 4 = 4
sum(Y) = 0 + 24 + 37 + 48 + 52 + 60 + 64 = 285
sum(X * Y) = (-5 * 0) + (-1 * 24) + (0 * 37) + (1 * 48) + (2 * 52) + (3 * 60) + (4 * 64) = 547
sum(X^2) = (-5)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 55
mean(X) = 4 / 7 ≈ 0.57
mean(Y) = 285 / 7 ≈ 40.71
Using these values, we can now calculate the slope (b) and y-intercept (a):
b = (7 * 547 - 4 * 285) / (7 * 55 - 4^2) ≈ 9.81
a = 40.71 - 9.81 * 0.57 ≈ 35.28
Therefore, the linear regression equation for the "full service" restaurant is approximately:
y = 9.81x + 35.28
wouldn't the correct answer be y=7.468x+53.766?
Apologies for the mistake in my previous response. You are correct, the correct linear regression equation for the "full service" restaurant is indeed:
y = 7.468x + 53.766
Thank you for pointing out the error.
To write a linear regression equation for the "full service" restaurant, we need to find the equation in the form of y = mx + b, where y represents the annual revenue in billions of dollars for the full-service restaurant, x represents the number of years since 1990, m represents the slope of the line, and b represents the y-intercept.
First, we need to find the slope (m) of the regression line. We can use the formula:
m = (Σxy - n(Σx)(Σy)) / (Σx^2 - n(Σx)^2)
To calculate this, we'll need the following sums:
Σxy = sum of the products of x and y for each year
Σx = sum of x (number of years since 1990)
Σy = sum of y (annual revenue)
Σx^2 = sum of the squares of x for each year
n = number of data points
Let's calculate these values first:
Σxy = (0*92) + (4*116) + (5*129) + (6*140) + (7*144) + (8*152) + (9*156) = 4040
Σx = 0 + 4 + 5 + 6 + 7 + 8 + 9 = 39
Σy = 92 + 116 + 129 + 140 + 144 + 152 + 156 = 1029
Σx^2 = (0^2) + (4^2) + (5^2) + (6^2) + (7^2) + (8^2) + (9^2) = 219
n = 7 (as there are 7 data points)
Now, we can substitute these values into the slope formula:
m = (Σxy - n(Σx)(Σy)) / (Σx^2 - n(Σx)^2)
= (4040 - 7(39)(1029)) / (219 - 7(39)^2)
= (4040 - 27693) / (219 - 6639)
= -23653 / -6420
= 3.68
Now that we have the slope (m), we can calculate the y-intercept (b) using the formula:
b = (Σy - m(Σx)) / n
Substituting the known values:
b = (1029 - 3.68(39)) / 7
= (1029 - 143.52) / 7
= 885.48 / 7
= 126.5
Therefore, the linear regression equation for the "full service" restaurant is:
y = 3.68x + 126.5
Note: Since the value of x represents the number of years since 1990, for example, x = 0 corresponds to 1990, x = 1 corresponds to 1991, and so on.