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 determine the regression equation, we can follow these steps:

Step 1: Calculate the mean (average) of x and y.
Step 2: Calculate the deviations of each x and y value from their respective means.
Step 3: Calculate the sum of the product of these deviations.
Step 4: Calculate the squares of the deviations of x and y.
Step 5: Calculate the square root of the sum of the squared deviations.
Step 6: Calculate the correlation coefficient, r.
Step 7: Determine the regression equation using the formula: y = a + bx.
Step 8: Determine the value of Y when X = 20, using the regression equation.

Let's go through each step.

Step 1:
Mean of x (x̄) = (2 + 5 + 6 + 8 + 9 + 11 + 15) / 7 = 8
Mean of y (ȳ) = (22 + 23 + 16 + 18 + 19 + 13 + 12) / 7 = 17

Step 2:
Deviation of x (dx) = x - x̄
Deviation of y (dy) = y - ȳ

dx = (2-8), (5-8), (6-8), (8-8), (9-8), (11-8), (15-8)
= -6, -3, -2, 0, 1, 3, 7

dy = (22-17), (23-17), (16-17), (18-17), (19-17), (13-17), (12-17)
= 5, 6, -1, 1, 2, -4, -5

Step 3:
Sum of the product of deviations (Σdxdy) = (-6×5) + (-3×6) + (-2×-1) + (0×1) + (1×2) + (3×-4) + (7×-5)
= -30 + (-18) + 2 + 0 + 2 + (-12) + (-35)
= -91

Step 4:
Squared deviations of x (dx^2) = dx × dx
Squared deviations of y (dy^2) = dy × dy

dx^2 = (-6)^2, (-3)^2, (-2)^2, (0)^2, (1)^2, (3)^2, (7)^2
= 36, 9, 4, 0, 1, 9, 49

dy^2 = (5)^2, (6)^2, (-1)^2, (1)^2, (2)^2, (-4)^2, (-5)^2
= 25, 36, 1, 1, 4, 16, 25

Step 5:
Square root of the sum of squared deviations (√Σdx^2) = √(36 + 9 + 4 + 0 + 1 + 9 + 49)
= √108
≈ 10.392

Step 6:
Correlation coefficient (r) = Σdxdy / (√Σdx^2 × √Σdy^2) = -91 / (10.392 × √108)
≈ -91 / (10.392 × 10.392)
≈ -91 / 107.981
≈ -0.843

So, the correlation coefficient, r, ≈ -0.843.

Step 7:
To determine the regression equation (y = a + bx), we need to calculate the values of a and b using the following formulas:

b = Σdxdy / Σdx^2 = -91 / 108
≈ -0.843

a = ȳ - b * x̄ = 17 - (-0.843) * 8
≈ 23.736

Therefore, the regression equation is: y = 23.736 - 0.843x.

Step 8:
To determine the value of Y when X = 20, we can substitute the value of x into the regression equation:
y = 23.736 - 0.843 * 20
= 23.736 - 16.86
≈ 6.876

Therefore, when X = 20, Y ≈ 6.876.

To calculate the correlation coefficient, regression equation, and determine the value of Y when X=20, we need to follow the steps below:

Step 1: Calculate the correlation coefficient, r.
The correlation coefficient, denoted by r, measures the strength and direction of the linear relationship between two variables. The formula to calculate r is as follows:

r = (Σxy - ((Σx)(Σy)/n)) / sqrt((Σx^2 - ((Σx)^2/n))(Σy^2 - ((Σy)^2/n)))

First, let's calculate the necessary values:

Σxy = (2 * 22) + (5 * 23) + (6 * 16) + (8 * 18) + (9 * 19) + (11 * 13) + (15 * 12)
Σxy = 44 + 115 + 96 + 144 + 171 + 143 + 180
Σxy = 893

Σx = 2 + 5 + 6 + 8 + 9 + 11 + 15
Σx = 56

Σy = 22 + 23 + 16 + 18 + 19 + 13 + 12
Σy = 123

Σx^2 = (2^2) + (5^2) + (6^2) + (8^2) + (9^2) + (11^2) + (15^2)
Σx^2 = 4 + 25 + 36 + 64 + 81 + 121 + 225
Σx^2 = 556

Σy^2 = (22^2) + (23^2) + (16^2) + (18^2) + (19^2) + (13^2) + (12^2)
Σy^2 = 484 + 529 + 256 + 324 + 361 + 169 + 144
Σy^2 = 2267

n = number of observations = 7

Substituting the values into the formula:

r = (Σxy - ((Σx)(Σy)/n)) / sqrt((Σx^2 - ((Σx)^2/n))(Σy^2 - ((Σy)^2/n)))
r = (893 - ((56 * 123)/7)) / sqrt((556 - ((56)^2/7))(2267 - ((123)^2/7)))
r = (893 - (6888/7)) / sqrt((556 - (3136/7))(2267 - (15129/7)))
r = (893 - 984) / sqrt((556 - 448)(2267 - 2161))
r = (-91) / sqrt(108 * 106)
r = -91 / sqrt(11448)
r ≈ -0.756

Therefore, the correlation coefficient, r ≈ -0.756.

Step 2: Determine the regression equation.
The regression equation allows us to predict the value of the dependent variable (Y) based on the independent variable (X). The equation has the form:

Y = a + (b * X)

where a is the intercept and b is the slope.

To calculate the intercept, a, and the slope, b, we can use the following formulas:

b = (nΣxy - (Σx)(Σy)) / (nΣx^2 - (Σx)^2)
a = (Σy - (b * Σx)) / n

Substituting the values:

b = (7 * 893 - (56 * 123)) / (7 * 556 - (56)^2)
b = (6251 - 6888) / (3892 - 3136)
b = -637 / 756
b ≈ -0.842

a = (123 - (-0.842 * 56)) / 7
a = (123 + 47.152) / 7
a ≈ 25.693

Therefore, the regression equation is Y = 25.693 - 0.842X.

Step 3: Determine the value of Y when X = 20.
To find the value of Y when X = 20, we substitute X = 20 into the regression equation:

Y = 25.693 - 0.842 * 20
Y = 25.693 - 16.84
Y ≈ 8.853

Therefore, when X = 20, Y ≈ 8.853.