1. Suppose that \(x\) and \(y\) are two variables of a population with \(x\) a predictor variable and \(y\) a response variable.

a. The distribution of all possible values of the response variable \(y\) corresponding to a particular value of the predictor variable \(x\) is called a distribution of the response variable.

b. State the four assumptions for regression inferences.

1. Suppose that $x$ and $y$ are two variables of a population with $x$ a predictor variable and $y$ a response variable.

a)

The predictor variable $\mathrm{x}$ is called the conditional distribution of the response variable.

b)

Assumptions for Regression inferences:

1. Population regression line: There are constants $\beta_{0}$ and $\beta_{1}$ such that, for each value $x$ of the predictor variable, the conditional mean of the response variable is $\beta_{0}+\beta_{1} x$.

2. Equal standard deviations: The conditional standard deviations of the response variable are the same for all values of the predictor variable. We denote this common standard deviation $\sigma$.

3. Normal populations: For each value of the predictor variable, the conditional distribution of the response variable is a normal distribution.

4. Independent observations: The observation of the response variable is independent of one another.