(correlation and regression)

for y = 25, what is the estimated value of value of x, from the following data:

X: 11 12 15 16 18 19 21

y: 21 15 13 12 11 10 9

a) 15 b)7 c)13.588 d) 14.986

To estimate the value of x when y is equal to 25 using correlation and regression, we need to perform the following steps:

Step 1: Calculate the correlation coefficient (r) between x and y. This will help us determine the strength and direction of the linear relationship between the two variables.

Step 2: Once we have the correlation coefficient, we can use it to calculate the regression equation (y = mx + b), which represents the line of best fit for the data.

Step 3: Substitute the given value of y (25) into the regression equation and solve for the corresponding value of x.

Let's go through each step in detail:

Step 1: Calculate the correlation coefficient (r)
To calculate the correlation coefficient, you can use a statistical software, a graphing calculator, or an online calculator. The correlation coefficient (r) ranges between -1 and 1. A value of -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.

For the given data, the correlation coefficient (r) is approximately -0.952.

Step 2: Calculate the regression equation (y = mx + b)
The regression equation represents the line that best fits the data. It can be calculated using the following formulas:

m = r * (Sy / Sx)
b = mean of y - (m * mean of x)

Substitute the values into the formulas:
m = -0.952 * (2.91 / 4.71) ≈ -0.587
b = 14.571 - (-0.587 * 15) ≈ 24.39

Therefore, the regression equation is: y = -0.587x + 24.39

Step 3: Substitute y = 25 into the regression equation and solve for x
25 = -0.587x + 24.39
0.587x = 25 - 24.39
0.587x ≈ 0.61
x ≈ 0.61 / 0.587 ≈ 1.04

Thus, the estimated value of x, when y is equal to 25, is approximately 1.04.

Therefore, none of the given options (a, b, c, or d) matches the estimated value.