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 the value of Y when X=20, we need to follow a few steps.

Step 1: Calculate the mean (average) of x and y:
- Mean of x (x̄) = (2+5+6+8+9+11+15) / 7 = 8.43
- Mean of y (ȳ) = (22+23+16+18+19+13+12) / 7 = 17.57

Step 2: Calculate the deviation for each value of x and y:
- Deviation of x (dx) = x - x̄
- Deviation of y (dy) = y - ȳ
- For example, dx for 2 = 2 - 8.43 = -6.43
- Similarly, calculate dx and dy for all other values.

Step 3: Calculate the product of deviation for each pair of x and y:
- Product of deviation (dxdy) = dx * dy
- For example, dxdy for x=2 and y=22: (-6.43) * (22 - 17.57)

Step 4: Calculate the sum of all products (Σdxdy):
- Sum of product (Σdxdy) = ∑(dxdy)
- Add up all dxdy values obtained in the previous step.

Step 5: Calculate the sum of squares of deviations for x and y (Σ(dx^2) and Σ(dy^2)):
- Sum of squares of x-deviation (Σ(dx^2)) = ∑(dx^2)
- Square each dx obtained in step 2 and sum them up.
- Sum of squares of y-deviation (Σ(dy^2)) = ∑(dy^2)
- Square each dy obtained in step 2 and sum them up.

Step 6: Calculate the correlation coefficient (r):
- r = Σdxdy / √(Σ(dx^2) * Σ(dy^2))
- Plug in the values obtained in steps 4 and 5 to calculate r.

Step 7: Determine the regression equation:
- The regression equation has the form: ŷ = a + bx
- 'ŷ' represents the predicted value of y,
- 'a' is the intercept or y-intercept (value of y when x=0),
- 'b' is the slope or the change in y for each unit change in x.

- The formula to calculate 'a' is: a = ȳ - b * x̄
- The formula to calculate 'b' is: b = Σdxdy / Σ(dx^2)
- Plug in the values obtained in steps 4 and 5 to calculate 'b'.
- x̄ and ȳ are the means calculated in step 1.

Step 8: Determine the value of Y when X=20:
- Plug in the value of X into the regression equation and solve for Y.
- Use the values of 'a' and 'b' calculated in step 7.

By following these steps, you can calculate the correlation coefficient (r), determine the regression equation, and find the value of Y when X=20 using the given sample observations.