Question

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.

Short answer

Part (a) conditional distribution of the response variable.

Part (b) Population regression line ,Equal standard deviation ,Normal populations ,Independent observations

Step-by-step solution

Step 1:Gven information 

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

Step 2:Explainaton Part (a)

a)

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

Step 3:Explaination Part (b)

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.