Decide if the following statements are true or false. Justify your answers.
b) If X and Y are independent random variables, then their coeﬂicient of correlation is zero.
c) Let X and Y be two random variables with zero coefficient of correlation. Then there is no linear association between X and Y.
d) If the correlation coefficient between X and Y is positive, then as X increases, Y tends to decrease.
e) See attached
f) See attached
g) In the simple linear regression model there are two parameters to be estimated.
h) See attached
i) In the simple linear regression model: see attached
j) If in a simple linear regression model r = 0.25 then 25% of the variation of y is explained by x.
k) In the simple linear regression model (see attached), we have (see attached).
The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
The Pearson coefficient of correlation between two variables X,Y is defined as
Let be a random variable with a symmetric distribution function P(x) about x=0 (say normal distribution) with zero mean:
Obviously X and Y are NOT independent, however, due to the symmetry of the distribution function we have an odd integrand when we calculate :
So the correlation is:
And we see that even if the correlation between X and Y is zero, it does not mean they are independent.
The solution decides if the given statements on linear statistical models are true or false.