Write a linear regression equation for a Full Service restaurant with the data 97; 118; 128; 140; 144; 147; 155. Fill in the correct answer for y = ________
To find the equation of a linear regression line, we need to find the slope (b) and the y-intercept (a).
First, we need the mean of the x-values (number of weeks, in this case) and the mean of the corresponding y-values (restaurant sales, in this case).
Let's calculate the mean of the x-values:
(97 + 118 + 128 + 140 + 144 + 147 + 155) / 7 = 125
Now, let's calculate the mean of the y-values:
(97 + 118 + 128 + 140 + 144 + 147 + 155) / 7 = 132
Next, we need to calculate the sum of the products of each x-value and y-value, and the sum of the squares of the x-values.
Sum of the products:
(97 * 97) + (118 * 118) + (128 * 128) + (140 * 140) + (144 * 144) + (147 * 147) + (155 * 155) = 155456
Sum of the squares of the x-values:
(97 * 97) + (118 * 118) + (128 * 128) + (140 * 140) + (144 * 144) + (147 * 147) + (155 * 155) = 16629
Now, let's calculate the slope (b):
b = (n * sum(xy) - sum(x) * sum(y)) / (n * sum(x^2) - (sum(x))^2)
b = (7 * 155456 - 7 * 125 * 132) / (7 * 16629 - (7 * 125)^2)
b = 2521 / 8694 ≈ 0.290
Finally, the y-intercept (a) can be calculated using the formula:
a = mean(y) - b * mean(x)
a = 132 - 0.290 * 125 = 132 - 36.25 ≈ 95.75
Therefore, the equation of the linear regression line is:
y = 0.290x + 95.75
So y = 0.290.
Now 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)
Certainly! Here's an expression using x as the variable:
5.678 * x + 3.214
Note: The coefficients in this expression are given as decimals rounded to the nearest thousandth.
To find the linear regression equation for the given data, we need to calculate the slope and the y-intercept. The general equation for a linear regression line is:
y = mx + b
Where:
- y represents the dependent variable (in this case, it is the revenue of the full-service restaurant).
- x represents the independent variable (in this case, it can be the number of months or any other relevant input that affects the revenue).
- m represents the slope of the line.
- b represents the y-intercept.
Let's calculate the slope and y-intercept using the given data:
x: 1 2 3 4 5 6 7
y: 97 118 128 140 144 147 155
First, calculate the mean (average) values for x and y:
mean(x) = (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7 = 4
mean(y) = (97 + 118 + 128 + 140 + 144 + 147 + 155) / 7 = 134
Next, calculate the differences from the mean for both x and y:
Δx = x - mean(x)
Δy = y - mean(y)
Δx: -3 -2 -1 0 1 2 3
Δy: -37 -16 -6 6 10 13 21
Now, calculate the sums of the products and the squares of the differences:
Σ(Δx*Δy) = (-3*-37) + (-2*-16) + (-1*-6) + (0*6) + (1*10) + (2*13) + (3*21) = -111 + 32 + 6 + 0 + 10 + 26 + 63 = 26
Σ(Δx^2) = (-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28
Now, let's calculate the slope (m):
m = Σ(Δx*Δy) / Σ(Δx^2) = 26 / 28 = 0.9286... (rounded to four decimal places)
Next, calculate the y-intercept (b):
b = mean(y) - m * mean(x) = 134 - 0.9286 * 4 = 134 - 3.7144... = 130.2857... (rounded to four decimal places)
Finally, substitute the values of m and b in the equation:
y = 0.9286x + 130.2857
So, the linear regression equation for the full-service restaurant is y = 0.9286x + 130.2857.
Therefore, the correct answer to fill in for y = _______ is:
y = 0.9286x + 130.2857