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.

Question

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
93
114
125
132
137
140
148
Fast Food
98
103
107
113
122
127
133
Question content area bottom
Part 1
a) Write a linear regression equation for the "full service" restaurant.
y=

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.)

Answer 1

To find the linear regression equation for the "full service" restaurant, we need to determine the equation of a line that best fits the given data. The equation of a line is typically expressed as y = mx + b, where m is the slope of the line and b is the y-intercept.

Using the given data, we can calculate the slope (m) and the y-intercept (b) by using the linear regression formula:

\[m = \frac{n \sum(xy) - (\sum x)(\sum y)} {n \sum(x^2) - (\sum x)^2}\]
\[b = \frac{\sum y - m(\sum x)} {n}\]

where n is the number of data points, \(\sum(x)\) is the sum of the x-values, \(\sum(y)\) is the sum of the y-values, \(\sum(xy)\) is the sum of the products of x and y values, and \(\sum(x^2)\) is the sum of the squares of the x-values.

Using the given data, we can calculate the following values:

n = 7 (since there are 7 data points)
\(\sum(x)\) = 7 + 9 + 10 + 11 + 12 + 13 + 14 = 76
\(\sum(y)\) = 93 + 114 + 125 + 132 + 137 + 140 + 148 = 889
\(\sum(xy)\) = (7)(93) + (9)(114) + (10)(125) + (11)(132) + (12)(137) + (13)(140) + (14)(148) = 8829
\(\sum(x^2)\) = (7^2) + (9^2) + (10^2) + (11^2) + (12^2) + (13^2) + (14^2) = 695

Now, let's calculate the slope (m) using the formula:

\(m = \frac{(7)(8829) - (76)(889)}{(7)(695) - (76)^2} \approx 2.924\)

Next, we can calculate the y-intercept (b) using the formula:

\(b = \frac{889 - (2.924)(76)}{7} \approx 41.018\)

Therefore, the linear regression equation for the "full service" restaurant is:

\[y ≈ 2.924x + 41.018\]

Note: Since we are dealing with revenue in billions of dollars, we can round the slope and y-intercept to three decimal places.