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 first need to find the mean (average) and standard deviation of both x and y values. Once we have these, we can proceed to calculate the correlation coefficient and regression equation.
a) Calculate the correlation coefficient (r):
Step 1: Find the mean of x and y.
Mean of x = (2 + 5 + 6 + 8 + 9 + 11 + 15) / 7 = 8
Mean of y = (22 + 23 + 16 + 18 + 19 + 13 + 12) / 7 = 17
Step 2: Calculate the deviation of each x and y value from their respective means.
Deviation of x values: -6, -3, -2, 0, 1, 3, 7
Deviation of y values: 5, 6, -1, 1, 2, -4, -5
Step 3: Calculate the product of the deviations for each pair of x and y.
Product of deviations: (-6 * 5) + (-3 * 6) + (-2 * -1) + (0 * 1) + (1 * 2) + (3 * -4) + (7 * -5) = -71
Step 4: Calculate the squared deviation for both x and y.
Squared deviation for x: (-6)^2 + (-3)^2 + (-2)^2 + 0^2 + 1^2 + 3^2 + 7^2 = 98
Squared deviation for y: (5)^2 + (6)^2 + (-1)^2 + (1)^2 + (2)^2 + (-4)^2 + (-5)^2 = 101
Step 5: Calculate the square root of the product of the squared deviations of x and y.
Square root of (98 * 101) = √9898 ≈ 99.49
Step 6: Calculate the correlation coefficient (r):
r = -71 / (7 * 99.49) ≈ -0.101
So, the correlation coefficient (r) is approximately -0.101.
b) Determine the regression equation:
The regression equation can be determined using the formula: y = mx + b, where m is the slope and b is the y-intercept.
Step 1: Calculate the slope (m):
m = r * (sy / sx)
where sy is the standard deviation of y and sx is the standard deviation of x.
Step 2: Calculate the y-intercept (b):
b = mean(y) - (m * mean(x))
Step 3: Substitute the values into the regression equation y = mx + b.
Using the calculated values:
m = -0.101 * (99.49 / √101) ≈ -0.306
b = 17 - (-0.306 * 8) ≈ 19.45
The regression equation is:
y ≈ -0.306x + 19.45
c) Determine the value of Y when X = 20:
Substitute the value of x (20) into the regression equation:
y = -0.306 * 20 + 19.45 ≈ 13.29
Therefore, when x = 20, the approximated value of y is 13.29.