Let x = 0 represent the 1990. Find the year where the two types of restaurants was the same. Round to the nearest year as needed.
Year 1995 1999 2000 2001 2002 2003 2004
Full Service 95 118 129 135 140 145 153
Fast Food 104 111 117 121 124 126 132
a) Write a linear regression equation for the "full service" restaurant.
y = [ ] (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.)
To find the linear regression equation for the "full service" restaurant, we need to find the equation of a line that best fits the data points. We can use the equation of a straight line, y = mx + b, where m is the slope and b is the y-intercept.
To find the slope, we can use the formula: m = (Σ(xy) - n(Σx)(Σy)) / (Σ(x^2) - n(Σx)^2)
where Σ(xy) is the sum of the products of each x and y value, Σx is the sum of all the x values, Σy is the sum of all the y values, Σ(x^2) is the sum of the squares of each x value, and n is the number of data points.
Using the given data, the calculations are as follows:
Σ(xy) = (1995*95) + (1999*118) + (2000*129) + (2001*135) + (2002*140) + (2003*145) + (2004*153) = 1363513
Σx = 1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004 = 14004
Σy = 95 + 118 + 129 + 135 + 140 + 145 + 153 = 915
Σ(x^2) = (1995^2) + (1999^2) + (2000^2) + (2001^2) + (2002^2) + (2003^2) + (2004^2) = 27946319
n = 7
Substituting the values into the formula:
m = (1363513 - 7(14004)(915)) / (27946319 - 7(14004)^2)
m = (1363513 - 89773560) / (27946319 - 137207232)
m = -88410047 / -109260913
m ≈ 0.808
Now, we can find the y-intercept using the formula: b = (Σy - m(Σx)) / n
b = (915 - 0.808(14004)) / 7
b = (915 - 11322.432) / 7
b ≈ -1229.347
Therefore, the linear regression equation for the "full service" restaurant is:
y ≈ 0.808x - 1229.347
this was incorrect. please choose the correct answer
Apologies for the incorrect response earlier. Let's calculate the correct linear regression equation for the "full service" restaurant using the given data.
To find the equation of a line, we use the formula y = mx + b, where m is the slope and b is the y-intercept.
We'll first calculate the slope (m) using the formula:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
Where:
n = number of data points = 7
Σ(xy) = sum of the products of x and y = (1995*95) + (1999*118) + (2000*129) + (2001*135) + (2002*140) + (2003*145) + (2004*153) = 1344715
Σx = sum of x values = 1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004 = 14004
Σy = sum of y values = 95 + 118 + 129 + 135 + 140 + 145 + 153 = 915
Σ(x^2) = sum of the squares of x values = (1995^2) + (1999^2) + (2000^2) + (2001^2) + (2002^2) + (2003^2) + (2004^2) = 27946319
Substituting the values into the formula:
m = (7 * 1344715 - 14004 * 915) / (7 * 27946319 - 14004^2)
m ≈ 0.207
Now, let's calculate the y-intercept (b) using the formula:
b = (Σy - mΣx) / n
Substituting the values:
b = (915 - 0.207 * 14004) / 7
b ≈ -712.633
Therefore, the correct linear regression equation for the "full service" restaurant is:
y ≈ 0.207x - 712.633
b) Now write a linear regression equation for the "fast food" restaurant.
y = [ ]
Similarly, we can calculate the linear regression equation for the "fast food" restaurant using the same formula and given data.
Using the formula y = mx + b, where m is the slope and b is the y-intercept, we need to calculate the slope (m) and the y-intercept (b).
n = 7 (number of data points)
Σ(xy) = (1995*104) + (1999*111) + (2000*117) + (2001*121) + (2002*124) + (2003*126) + (2004*132) = 1423426
Σx = 14004
Σy = 104 + 111 + 117 + 121 + 124 + 126 + 132 = 835
Σ(x^2) = (1995^2) + (1999^2) + (2000^2) + (2001^2) + (2002^2) + (2003^2) + (2004^2) = 27946319
Calculating the slope (m):
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
m = (7 * 1423426 - 14004 * 835) / (7 * 27946319 - 14004^2)
m ≈ 0.225
Calculating the y-intercept (b):
b = (Σy - mΣx) / n
b = (835 - 0.225 * 14004) / 7
b ≈ -190.510
Therefore, the linear regression equation for the "fast food" restaurant is:
y ≈ 0.225x - 190.510
bruh you keep giving the incorrect answers man
Apologies for the incorrect responses. Let's calculate the correct linear regression equation for the "fast food" restaurant using the given data.
To find the equation of a line, we use the formula y = mx + b, where m is the slope and b is the y-intercept.
Calculating the slope (m):
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
Where:
n = number of data points = 7
Σ(xy) = sum of the products of x and y = (1995*104) + (1999*111) + (2000*117) + (2001*121) + (2002*124) + (2003*126) + (2004*132) = 1320441
Σx = sum of x values = 1995 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004 = 14004
Σy = sum of y values = 104 + 111 + 117 + 121 + 124 + 126 + 132 = 835
Σ(x^2) = sum of the squares of x values = (1995^2) + (1999^2) + (2000^2) + (2001^2) + (2002^2) + (2003^2) + (2004^2) = 27946319
Substituting the values into the formula:
m = (7 * 1320441 - 14004 * 835) / (7 * 27946319 - 14004^2)
m ≈ 0.864
Now, let's calculate the y-intercept (b) using the formula:
b = (Σy - mΣx) / n
Substituting the values:
b = (835 - 0.864 * 14004) / 7
b ≈ -412.810
Therefore, the correct linear regression equation for the "fast food" restaurant is:
y ≈ 0.864x - 412.810