Question

1. For any given X, what is Y-hat?

In other words, what is probably the best way to describe Y-hat in terms of X. (This is looking for a definition, please don't just say"predicted value)

2.If the regression line and the mean of Y line are the same line, what does this mean? (This is ask for observation too)

3.I want to test to see if a die is fair. Why shouldn’t I just do one test to see if there are too many ones, one test to see if there are too many twos, one test to see if there are too many threes, etc.?

Answer 1

From the given data,

1)

Y-hat is dependent variable is also known as response variable which we put on y-axis on graph. It is depend on the value of x which is explanatory variable( independent variable). It varies  according to value of x.
Y-hat = a + b*x

x= (Y-hat-a)/b . where b is slope and a is intercept.

2)

Regression line of Y on X is

Y=E(Y/X) and mean of Y means E(Y) or expected value of Y.if they are same we can say that

E(Y)=E(Y/X)..this is only possible when X and Y are independent.hence this means the two variables related to these are independent

3)

First we need to understand the meaning of fair die.Fair die means each of the faces having equal or same probability of landing on that face. Each having equal chance to occur. Therefore, the probability of each face is 1/6.

Now, need to understand about the loaded die. The loaded die is designed in a special way that 3 or 4 are relatively likely to occur and other faces are unlikely to occur. Therefore, it does not have equal chance which leading unequal probability of each face. Hence, we should not test to see if there are too many one's, one test to see if there are too many two's, one test to see if there are too many three's and so on.