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), regression equation, and find the value of Y when X=20, we will follow these steps:
Step 1: Find the mean of x (x̄) and y (ȳ).
To calculate the mean, sum up all the values and divide by the total number of observations.
For x: x̄ = (2 + 5 + 6 + 8 + 9 + 11 + 15) / 7 = 7.714
For y: ȳ = (22 + 23 + 16 + 18 + 19 + 13 + 12) / 7 = 17.571
Step 2: Calculate the differences between each value of x and the mean of x (x - x̄), and y and the mean of y (y - ȳ).
x: -5.714, -2.714, -1.714, 0.286, 1.286, 3.286, 7.286
y: 4.429, 5.429, -1.571, 0.429, 1.429, -4.571, -5.571
Step 3: Calculate the product of the differences, (x - x̄)(y - ȳ), for each pair of values and sum them up.
Product of differences: (-5.714 * 4.429) + (-2.714 * 5.429) + (-1.714 * -1.571) + (0.286 * 0.429) + (1.286 * 1.429) + (3.286 * -4.571) + (7.286 * -5.571) = -70.293
Step 4: Calculate the sum of the squares of the differences for x and y.
Sum of squares of differences:
For x: (-5.714)^2 + (-2.714)^2 + (-1.714)^2 + (0.286)^2 + (1.286)^2 + (3.286)^2 + (7.286)^2 = 140.571
For y: (4.429)^2 + (5.429)^2 + (-1.571)^2 + (0.429)^2 + (1.429)^2 + (-4.571)^2 + (-5.571)^2 = 84.571
Step 5: Calculate the correlation coefficient (r) using the formula:
r = ∑((x - x̄)(y - ȳ)) / √(∑(x - x̄)^2 * ∑(y - ȳ)^2)
r = -70.293 / √(140.571 * 84.571) = -70.293/ √11881.157 = -70.293/ 108.968 ≈ -0.645
Step 6: Determine the regression equation.
The regression equation is in the form Y = a + bx, where a is the y-intercept and b is the slope.
First, calculate the slope (b):
b = r * (Sy / Sx)
Where Sy is the standard deviation of y and Sx is the standard deviation of x.
Sy = √(∑(y - ȳ)^2 / (n - 1))
Sy = √(84.571 / 6) = √14.095 ≈ 3.751
Sx = √(∑(x - x̄)^2 / (n - 1))
Sx = √(140.571 / 6) = √23.428 ≈ 4.84
b = -0.645 * (3.751 / 4.84) ≈ -0.499
Next, calculate the y-intercept (a):
a = ȳ - b * x̄
a = 17.571 - (-0.499 * 7.714) ≈ 21.774
Therefore, the regression equation is:
Y ≈ 21.774 - 0.499x
Step 7: Determine the value of Y when X = 20.
Plug X = 20 into the regression equation and solve for Y:
Y = 21.774 - 0.499 * 20
Y ≈ 21.774 - 9.98
Y ≈ 11.794
Therefore, when X = 20, Y is approximately 11.794.