Calculate a linear regression, the residuals, and the correlation between two variables using the following info: number of people:12,14,15,18,20,16,14,13,18,19,20,22 and
cost of drinks:24,30,36,38,65,44,36,30,39,76,80,85
Using an online calculator, I have the following:
12 data pairs (x,y):
( 12.0 , 24.0 ); ( 14.0 , 30.0 ); ( 15.0 , 36.0 ); ( 18.0 , 38.0 ); ( 20.0 , 65.0 ); ( 16.0 , 44.0 ); ( 14.0 , 36.0 ); ( 13.0 , 30.0 ); ( 18.0 , 39.0 ); ( 19.0 , 76.0 ); ( 20.0 , 80.0 ); ( 22.0 , 85.0 );
Regression equation:
predicted y = -53 + 6.06x
Correlation: r = 0.894
To find residuals:
Y(hat) comes from substituting an x value into a regression equation and solving for y(hat). Y(hat) is also called the predicted y value in a regression equation.
Let's use an example. Suppose the regression equation is this:
y(hat) = 2.75 + .5x
If x = 1, then y(hat) = 3.25
Suppose y = 3. Then y - y(hat) would be 3 - 3.25 = -0.25 (using the above example). This would be your residual.
The residuals in regression are measuring how far each observed y is from the regression line, y(hat), for a given value of x.
I hope this helps.
To calculate a linear regression, residuals, and correlation between the number of people and the cost of drinks, follow these steps:
1. Create a table to organize the data:
Number of People (x) | Cost of Drinks (y)
------------------- | -----------------
12 | 24
14 | 30
15 | 36
18 | 38
20 | 65
16 | 44
14 | 36
13 | 30
18 | 39
19 | 76
20 | 80
22 | 85
2. Calculate the mean (average) of both variables:
Mean of x (number of people):
mean_x = (12 + 14 + 15 + 18 + 20 + 16 + 14 + 13 + 18 + 19 + 20 + 22) / 12
mean_x = 199 / 12
mean_x = 16.58 (rounded to 2 decimal places)
Mean of y (cost of drinks):
mean_y = (24 + 30 + 36 + 38 + 65 + 44 + 36 + 30 + 39 + 76 + 80 + 85) / 12
mean_y = 563 / 12
mean_y = 46.92 (rounded to 2 decimal places)
3. Calculate the sum of squares of both variables:
Sum of squares of x (number of people):
SS_x = (12 - 16.58)^2 + (14 - 16.58)^2 + (15 - 16.58)^2 + (18 - 16.58)^2 + (20 - 16.58)^2 + (16 - 16.58)^2 + (14 - 16.58)^2 + (13 - 16.58)^2 + (18 - 16.58)^2 + (19 - 16.58)^2 + (20 - 16.58)^2 + (22 - 16.58)^2
SS_x = 172.64 (rounded to 2 decimal places)
Sum of squares of y (cost of drinks):
SS_y = (24 - 46.92)^2 + (30 - 46.92)^2 + (36 - 46.92)^2 + (38 - 46.92)^2 + (65 - 46.92)^2 + (44 - 46.92)^2 + (36 - 46.92)^2 + (30 - 46.92)^2 + (39 - 46.92)^2 + (76 - 46.92)^2 + (80 - 46.92)^2 + (85 - 46.92)^2
SS_y = 3934.92 (rounded to 2 decimal places)
4. Calculate the sum of products:
Sum of products (xy):
sum_xy = (12 * 24) + (14 * 30) + (15 * 36) + (18 * 38) + (20 * 65) + (16 * 44) + (14 * 36) + (13 * 30) + (18 * 39) + (19 * 76) + (20 * 80) + (22 * 85)
sum_xy = 25440
5. Calculate the slope (b1):
b1 = sum_xy / SS_x
b1 = 25440 / 172.64
b1 = 147.41 (rounded to 2 decimal places)
6. Calculate the y-intercept (b0):
b0 = mean_y - (b1 * mean_x)
b0 = 46.92 - (147.41 * 16.58)
b0 = -2290.88 (rounded to 2 decimal places)
7. Write the linear regression equation:
The linear regression equation is:
y = -2290.88 + 147.41x
8. Calculate the residuals:
To calculate the residuals for each observation, substitute the x value into the linear regression equation and subtract the corresponding y value. The difference will be the residual.
For example:
Residual for the first observation (12, 24):
Residual = 24 - (-2290.88 + 147.41 * 12)
Residual = 24 + 617.08 = 641.08
Repeat this calculation for each observation to find the residual for all data points.
9. Calculate the correlation (r):
r = (n * sum_xy - sum_x * sum_y) / sqrt((n * SS_x - sum_x^2) * (n * SS_y - sum_y^2))
Where:
n = number of observations (12)
sum_x = sum of x values
sum_y = sum of y values
sum_x = 199
sum_y = 563
r = (12 * 25440 - 199 * 563) / sqrt((12 * 172.64 - 199^2) * (12 * 3934.92 - 563^2))
r = 167760 / sqrt((2071.68 - 39601) * (47219.04 - 316969))
r = 167760 / sqrt((-37529.32) * (-269749.92))
r = 167760 / sqrt(10114404682.94)
r = 167760 / 100571.78
r = 1.67 (rounded to 2 decimal places)
The correlation (r) between the number of people and the cost of drinks is approximately 1.67.
To calculate the linear regression, residuals, and correlation between the number of people and the cost of drinks, you can follow these steps:
1. First, create a table of the given data:
Number of People: 12, 14, 15, 18, 20, 16, 14, 13, 18, 19, 20, 22
Cost of Drinks: 24, 30, 36, 38, 65, 44, 36, 30, 39, 76, 80, 85
2. Calculate the mean (average) of the number of people and the cost of drinks. Let's denote them as num_mean and cost_mean, respectively.
num_mean = (12 + 14 + 15 + 18 + 20 + 16 + 14 + 13 + 18 + 19 + 20 + 22) / 12
= 193 / 12
= 16.08 (rounded to two decimal places)
cost_mean = (24 + 30 + 36 + 38 + 65 + 44 + 36 + 30 + 39 + 76 + 80 + 85) / 12
= 563 / 12
= 46.92 (rounded to two decimal places)
3. Calculate the deviations from the mean for both variables (number of people and cost of drinks) and create a new table:
Number of People: X | Deviation from Mean (X - num_mean) | Cost of Drinks: Y | Deviation from Mean (Y - cost_mean)
-----|---------------------------------------| -----|-------------------------------------------
12 | 12 - 16.08 | 24 | 24 - 46.92
14 | 14 - 16.08 | 30 | 30 - 46.92
15 | 15 - 16.08 | 36 | 36 - 46.92
18 | 18 - 16.08 | 38 | 38 - 46.92
20 | 20 - 16.08 | 65 | 65 - 46.92
16 | 16 - 16.08 | 44 | 44 - 46.92
14 | 14 - 16.08 | 36 | 36 - 46.92
13 | 13 - 16.08 | 30 | 30 - 46.92
18 | 18 - 16.08 | 39 | 39 - 46.92
19 | 19 - 16.08 | 76 | 76 - 46.92
20 | 20 - 16.08 | 80 | 80 - 46.92
22 | 22 - 16.08 | 85 | 85 - 46.92
4. Calculate the product of the deviations (X - num_mean) * (Y - cost_mean) for each row and add them up:
Sum of (X - num_mean) * (Y - cost_mean) = (12 - 16.08) * (24 - 46.92) +
(14 - 16.08) * (30 - 46.92) +
(15 - 16.08) * (36 - 46.92) +
(18 - 16.08) * (38 - 46.92) +
(20 - 16.08) * (65 - 46.92) +
(16 - 16.08) * (44 - 46.92) +
(14 - 16.08) * (36 - 46.92) +
(13 - 16.08) * (30 - 46.92) +
(18 - 16.08) * (39 - 46.92) +
(19 - 16.08) * (76 - 46.92) +
(20 - 16.08) * (80 - 46.92) +
(22 - 16.08) * (85 - 46.92)
= (-4.08) * (-22.92) +
(-2.08) * (-16.92) +
(-1.08) * (-10.92) +
(1.92) * (-8.92) +
(3.92) * (18.08) +
(-0.08) * (-2.92) +
(-2.08) * (-10.92) +
(-3.08) * (-16.92) +
(1.92) * (-7.92) +
(2.92) * (29.08) +
(3.92) * (33.08) +
(5.92) * (38.08)
= 10295.22
5. Calculate the sum of squares of deviations of X (Number of People) and Y (Cost of Drinks):
Sum of (X - num_mean)^2 = (12 - 16.08)^2 + (14 - 16.08)^2 + (15 - 16.08)^2 + (18 - 16.08)^2 + (20 - 16.08)^2 +
(16 - 16.08)^2 + (14 - 16.08)^2 + (13 - 16.08)^2 + (18 - 16.08)^2 + (19 - 16.08)^2 +
(20 - 16.08)^2 + (22 - 16.08)^2
= 27.18
Sum of (Y - cost_mean)^2 = (24 - 46.92)^2 + (30 - 46.92)^2 + (36 - 46.92)^2 + (38 - 46.92)^2 + (65 - 46.92)^2 +
(44 - 46.92)^2 + (36 - 46.92)^2 + (30 - 46.92)^2 + (39 - 46.92)^2 + (76 - 46.92)^2 +
(80 - 46.92)^2 + (85 - 46.92)^2
= 11651.68
6. Calculate the slope (b) of the linear regression line:
b = Sum of (X - num_mean) * (Y - cost_mean) / Sum of (X - num_mean)^2
= 10295.22 / 27.18
= 379.37 (rounded to two decimal places)
7. Calculate the y-intercept (a) of the linear regression line:
a = cost_mean - (b * num_mean)
= 46.92 - (379.37 * 16.08)
= -5988.71 (rounded to two decimal places)
8. The linear regression equation will be in the form of:
cost_of_drinks = a + b * number_of_people
Substituting the values of a and b, we get:
cost_of_drinks = -5988.71 + 379.37 * number_of_people
9. Calculate the residuals (differences between actual data points and predicted values):
For each data point (X, Y), calculate Y_pred using the linear regression equation, and then calculate the difference (residual).
For example, let's calculate the residual for the first data point (12, 24):
Y_pred = -5988.71 + 379.37 * 12
= -5988.71 + 4552.44
= -1436.27 (rounded to two decimal places)
Residual = 24 - (-1436.27)
= 1460.27 (rounded to two decimal places)
Calculate residuals for each data point in a similar manner.
10. Finally, to calculate the correlation between the two variables, divide the sum of (X - num_mean) * (Y - cost_mean) by the square root of the product of the sum of (X - num_mean)^2 and the sum of (Y - cost_mean)^2.
correlation = (Sum of (X - num_mean) * (Y - cost_mean)) / sqrt((Sum of (X - num_mean)^2) * (Sum of (Y - cost_mean)^2))
= 10295.22 / sqrt(27.18 * 11651.68)
= 0.9064 (rounded to four decimal places)
Therefore, the linear regression equation is cost_of_drinks = -5988.71 + 379.37 * number_of_people, the residuals can be calculated for each data point, and the correlation between the number of people and the cost of drinks is 0.9064.