Please help me understand these concepts

1. Teach me How do I draw an approximation to the best fit line?
2. Teach me How do I know whether the correlation is closest to -1, -0.5, 0, 0.5, or 1
3. Teach me How do I know what to expect the sign for the coefficient of X²
   

4. The scatter plot below between Y and X was generated by first creating a variable U that took on the values (0.04, 0.08, 0.12, ... , 0.96, 1.00), then creating X = cos(RU) and Y = sin(TU) + & where "cos" is the trigonometric cosine function, "sin" is the trigonometric sine function, & is a normally distributed random variable with variance 1/100, and n is the well-known constant relating the circumference to the diameter of a circle (i.e., n = 3.14159....). 1 . 0 0 .9 - 0 . 8 0 . 7 7 0 . 6- 0 . 5- 0.4 0. 3 7 0 . 2- 0. 17 - 1 .0 - 0 . 8 - 0 . 6 -0. 2 0. 2 0 . 6 0. 8 1.0 After fitting the regression of Y on X, the residual plot looks like the following: 0 . 8 0 . 6- 0. 47 J. 2 Residual 0. 2 + + + -D . 4- 0 . 67 -0. 8 - 1 . 1 0 . 5 0 . 1. 0 12 R T Y U O P G H J K4.(a) (5 points) On the scatterplot of Y versus X, draw an approximation to the best-fit line. 4.(b) (5 points) Comment on whether you think the sample correlation between Y and X is closest to -1, -0.5, 0, 0.5, or 1, and explain your reasoning in a sentence. 4.(c) (5 points) From the residual plot, it appears that there is non-linearity in the data. Suppose we were to regress Y on two variables, X and X'. Would you expect the sign for the coefficient of X' to be positive or negative? Explain your reasoning. 13 S 8 T 4 5 6 7 8 9 O R T Y U O P