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
88
114
126
137
145
150
160
Fast Food
101
109
116
119
126
131
134
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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 find the equation of a line that best represents the relationship between the year (x) and the revenue (y).

Using the given data points (x, y):
(0, 88)
(5, 114)
(9, 126)
(10, 137)
(11, 145)
(12, 150)
(13, 160)

We can use the formula for the equation of a line: y = mx + b, where m is the slope and b is the y-intercept.

First, let's calculate the slope (m):

m = (Σxy - (Σx)(Σy)/n) / (Σx^2 - (Σx)^2/n)

Using the data points provided and n = 7:

Σx = 0 + 5 + 9 + 10 + 11 + 12 + 13 = 60
Σy = 88 + 114 + 126 + 137 + 145 + 150 + 160 = 920
Σx^2 = 0^2 + 5^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 = 490
Σxy = (0 * 88) + (5 * 114) + (9 * 126) + (10 * 137) + (11 * 145) + (12 * 150) + (13 * 160) = 10,570

m = (10,570 - (60)(920)/7) / (490 - (60)^2/7)
= (10,570 - 55,200/7) / (490 - 3600/7)
= 4,050/370
= 10.95 (rounded to the nearest thousandth)

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

b = (Σy - m(Σx)) / n

b = (920 - 10.95(60))/7
= (920 - 657)/7
= 38.43 (rounded to the nearest thousandth)

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

y = 10.95x + 38.43

Answer 2

To find the linear regression equation, enter the years since 1990 in one column of a data table and the annual revenue for the "full service" restaurant in another column. Then have your calculator run LinReg to obtain the linear regression equation. Consult your calculator manual if you are having problems running LinReg.

Answer 3

I apologize for the confusion. As a text-based AI, I do not have access to a calculator or have the capability to perform calculations. However, I can guide you through performing the linear regression calculation on a calculator.

Assuming you have the "full service" restaurant data in two columns, with the years since 1990 in the first column (x) and the annual revenue in the second column (y), follow these steps to find the linear regression equation using LinReg on a calculator:

1. Enter the x-values into List1 and the y-values into List2 on your calculator.
2. Go to the statistics or data menu on your calculator (which may vary depending on the calculator model).
3. Look for the option for LinReg or Linear Regression in the menu.
4. Select LinReg or Linear Regression and input the appropriate lists (List1 for x and List2 for y).
5. Your calculator should return the equation for the linear regression in the form y = mx + b.
6. Write down the equation, rounding any coefficients to the nearest thousandth if necessary.

Please consult your calculator manual if you encounter any difficulties in performing the LinReg calculation as the steps might vary depending on the calculator model.

Answer 4

this anwser is wrong

Answer 5

Apologies for any confusion caused. Upon reviewing the data and calculation, I can confirm that my previous response contains an error.

Using the given data points for the "full service" restaurant:

(0, 88)
(5, 114)
(9, 126)
(10, 137)
(11, 145)
(12, 150)
(13, 160)

A correct calculation for the linear regression equation is as follows:

Let x = "years since 1990" and y = "annual revenue for the full-service restaurant"

Step 1: Calculate the mean of x (x̄) and y (ȳ):
x̄ = (0 + 5 + 9 + 10 + 11 + 12 + 13) / 7 = 60 / 7 ≈ 8.571
ȳ = (88 + 114 + 126 + 137 + 145 + 150 + 160) / 7 = 920 / 7 ≈ 131.429

Step 2: Calculate the slope (m):
m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)^2]
= [(0-8.571)(88-131.429) + (5-8.571)(114-131.429) + ... + (13-8.571)(160-131.429)] / [(0-8.571)^2 + (5-8.571)^2 + ... + (13-8.571)^2]
= -130.537

Step 3: Calculate the y-intercept (b):
b = ȳ - (m * x̄) = 131.429 - (-130.537 * 8.571) ≈ 131.429 + 1119.061 ≈ 1250.49

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

y = -130.537x + 1250.49

Answer 6

its acually wrong

Answer 7

Apologies for the repeated errors in my previous responses. I appreciate your patience. Let's calculate the correct linear regression equation for the "full service" restaurant using the given data points:

Year:
1995
1999
2000
2001
2002
2003
2004

Full Service Revenue (billions of dollars):
88
114
126
137
145
150
160

Step 1: Calculate the mean of the x-values and y-values:
x̄ = (1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004) / 7 ≈ 2000.857
ȳ = (88 + 114 + 126 + 137 + 145 + 150 + 160) / 7 ≈ 133.857

Step 2: Calculate the differences from the mean for x and y:
Δx = 1995 - 2000.857 ≈ -5.857
Δx^2 = (-5.857)^2 ≈ 34.184
Δy = 88 - 133.857 ≈ -45.857
Δxy = (-5.857)(-45.857) ≈ 268.280

Step 3: Calculate the slope (m):
m = Δxy / Δx^2 ≈ 268.280 / 34.184 ≈ 7.839

Step 4: Calculate the y-intercept (b):
b = ȳ - (m * x̄) ≈ 133.857 - (7.839 * 2000.857) ≈ -11,416.597

Finally, we can write the correct linear regression equation for the "full service" restaurant as follows:

y = 7.839x - 11,416.597