the following sample observations were randomly selected:
x: 2,5,6,8,9,11,15
y: 22,23,16,18,19,13,12
a)calculate the correlation coefficient, r
b) determine the regression equation
c) determine the value of Y when X=20
To calculate the correlation coefficient (r) and regression equation for the given data, follow these steps:
Step 1: Calculate the means of x and y.
x: 2, 5, 6, 8, 9, 11, 15
y: 22, 23, 16, 18, 19, 13, 12
Mean of x (x̄) = (2 + 5 + 6 + 8 + 9 + 11 + 15) / 7 = 8.857
Mean of y (ȳ) = (22 + 23 + 16 + 18 + 19 + 13 + 12) / 7 = 17.714
Step 2: Calculate the deviations from the means for x and y.
x: -6.857, -3.857, -2.857, -0.857, 0.143, 2.143, 6.143
y: 4.286, 5.286, -1.714, 0.286, 1.286, -4.714, -5.714
Step 3: Calculate the product of the deviations for each pair of x and y.
Product of deviations = (-6.857 * 4.286) + (-3.857 * 5.286) + (-2.857 * -1.714) + (-0.857 * 0.286) + (0.143 * 1.286) + (2.143 * -4.714) + (6.143 * -5.714) = -43.714
Step 4: Calculate the sum of the squared deviations for x and y.
Sum of squared deviations for x = (-6.857)^2 + (-3.857)^2 + (-2.857)^2 + (-0.857)^2 + (0.143)^2 + (2.143)^2 + (6.143)^2 = 144.857
Sum of squared deviations for y = (4.286)^2 + (5.286)^2 + (-1.714)^2 + (0.286)^2 + (1.286)^2 + (-4.714)^2 + (-5.714)^2 = 127.857
Step 5: Calculate the square root of the product of the sum of squared deviations for x and y.
√(Sum of squared deviations for x * Sum of squared deviations for y) = √(144.857 * 127.857) ≈ 141.034
Step 6: Calculate the correlation coefficient (r) using the formula:
r = Product of deviations / √(Sum of squared deviations for x * Sum of squared deviations for y)
r = -43.714 / 141.034 ≈ -0.310
Therefore, the correlation coefficient (r) for the given data is approximately -0.310.
Now, to determine the regression equation:
Step 7: Calculate the slope (b) of the regression line.
b = r * (Standard deviation of y / Standard deviation of x)
Step 8: Calculate the standard deviation of x and y.
Standard deviation of x = √(((-6.857)^2 + (-3.857)^2 + (-2.857)^2 + (-0.857)^2 + (0.143)^2 + (2.143)^2 + (6.143)^2) / (7 - 1)) ≈ 4.416
Standard deviation of y = √(((4.286)^2 + (5.286)^2 + (-1.714)^2 + (0.286)^2 + (1.286)^2 + (-4.714)^2 + (-5.714)^2) / (7 - 1)) ≈ 3.573
Step 9: Calculate the slope (b).
b = -0.310 * (3.573 / 4.416) ≈ -0.251
Step 10: Calculate the y-intercept (a) of the regression line.
a = ȳ - (b * x̄)
a = 17.714 - (-0.251 * 8.857) ≈ 20.925
Step 11: Write the regression equation.
The regression equation is y = -0.251x + 20.925.
Finally, to determine the value of y when x = 20:
Step 12: Substitute x = 20 into the regression equation.
y = -0.251(20) + 20.925
y ≈ 15.475
Therefore, when x = 20, the value of y is approximately 15.475.
To calculate the correlation coefficient (r), the regression equation, and the value of Y when X=20, we can follow these steps:
a) Calculate the correlation coefficient, r:
The correlation coefficient measures the linear relationship between two variables. We can use the formula:
r = (∑((X-X̄)(Y-Ȳ))) / (sqrt(∑(X-X̄)²) * sqrt(∑(Y-Ȳ)²))
Where:
- X and Y are the given variables.
- X̄ and Ȳ are the means of X and Y, respectively.
- ∑ represents the summation, summing over all the given values.
Calculations for the numerator:
(X-X̄) = (2-9.57), (5-9.57), (6-9.57), (8-9.57), (9-9.57), (11-9.57), (15-9.57)
(Y-Ȳ) = (22-16.71), (23-16.71), (16-16.71), (18-16.71), (19-16.71), (13-16.71), (12-16.71)
Multiply each pair together and find the sum of these products.
Calculations for the denominator:
(X-X̄)² = (2-9.57)², (5-9.57)², (6-9.57)², (8-9.57)², (9-9.57)², (11-9.57)², (15-9.57)²
(Y-Ȳ)² = (22-16.71)², (23-16.71)², (16-16.71)², (18-16.71)², (19-16.71)², (13-16.71)², (12-16.71)²
Sum the squares of (X-X̄) and (Y-Ȳ).
Using these values, we can calculate the correlation coefficient (r).
b) Determine the regression equation:
The regression equation represents the relationship between X and Y. We can use the formula:
Y = a + bX
Where:
- Y is the dependent variable.
- X is the independent variable.
- a is the y-intercept (where the regression line crosses the y-axis).
- b is the slope of the regression line.
The values of a and b can be calculated using the formulas:
b = (∑(X- X̄)(Y- Ȳ)) / ∑(X- X̄)²
a = Ȳ - bX̄
Calculate the means of X and Y:
X̄ = (∑X) / n
Ȳ = (∑Y) / n
Substitute these values into the formulas to find the values of a and b.
c) Determine the value of Y when X=20:
Using the regression equation Y = a + bX, substitute X=20 into the equation and solve for Y.
By following these steps, we can calculate the correlation coefficient (r), determine the regression equation, and find the value of Y when X=20 for the given data set.