I'd an exam recently and I got 2 exercises from regression part of an exam I couldn't have solved. Can you help me. I will provide both exercises and my attempts to solving them. Exercises:
- Based on a random sample of traits X and Y, the following results were obtained: x = y = 2 (mean), s(x) = 1, s(y) = 0.5, cov(x,y) = 0.4. Determine the equation of the simple regression line for trait Y with respect to trait X and estimate its goodness of fit to the empirical data, knowing that the random sample consisted of 250 elements.
- We have the following multiple regression model: y = 2 + 0.4x1 - 2.5x2 + 0.4x3. Which of the explanatory variables has the greatest impact on the dependent variable? What is the value of the multiple correlation coefficient if it is known that this model is fitted to the empirical data at a 95% confidence level?
My attempt to solve it:
y = ax + b a = (cov(x,y))/(s(x)^2) = 0.4
Now I attempted to calculate b from the equation above, but how can I do that if I do not have values of x and y? I tried to insert the means here, so I end with: 2 = 0.4*2 + b => b = 1.2
The equation is: y = 0.4x + 1.2
Is this correct? How to calculate the rest of the exercise?Didn't know at all how to calculate it so I use common sense and written such justification:
'x1 and x3 have equal coefficients, so their impact will be the same. x2 has the highest absolute coefficient value, so x2 will have the greatest influence on the value of y.'
To 'prove' it additionally I've written a table with changing x1, x2 and x3 values:
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
[0, 0, 2]
[0, 2, 0]
[2, 0, 0]
And then calculate y values for each vector to show that the change of x2 has most significant effect on y (and same values of x1 and x3 result in same results). Is such justification acceptable or need it additional calculations? How to calculate the rest of the exercise?
Thank's for help in advance. Please provide missing formulas to make it more readable.