Prove that the least-squares t in simple-regression analysis has the following properties: Zgii = 0 2- 2091' a)(i - ) = Zdz' - Q) : 0 Suppose that...

Hi, please help me with the following prove questions. If it is easier, please return in hand written.

Prove that the least-squares fit in simple-regression analysis has the following properties:1. Zgiéi = 02- 2091‘ — fia)(fli - fl) = Zédfiz‘ - Q) : 0Suppose that the means and standard deviations of y and a: are the same: i; : 5E and sd('y) : sd(a:). 1. Show that, under these circumstances (Byte : flxly : Try where 5y|m is the least-squares slope for the simple regression of 3; on 3:, 5me is the least-squares slopefor the simple regression of a: on y, and army is the correlation between the two variables. Show that theintercepts are also the same, aylm : amly. 2. Why, if aylr : (my and 61m, : fiwlyv is the least squares line for the regression of y on a: different fromthe line for the regression of a: on y (when my < l)? 3. Imagine that educational researchers wish to assess the efl'icacy of a new program to improve hereading performance of children. To test the program, they recruit a group of children who are readingsubstantially vbelow grade level; after a year in the program, the researchers observe that the children,on average, have imporved their reading performance. Why is this a weak research design? How couldit be improved?