**The regression equation is the line that best fits a set of data as determined by having the least squared error.****The slope shows the amount of change in Y for a one-unit****increase in X.****After conducting a hypothesis test and the slope of the regression equation is nonzero, you can then conclude that your predictor variable, X, causes Y.**

The question aims to find the correct statements about regression with one predictor variable, which is also commonly referred to as Simple Regression.

Simple Regression is a statistical tool used to determine the relationship between one dependent and one independent variable based on the given observations. The linear regression model can be expressed as the following equation:

A simple regression model particularly refers to the modeling between only one dependent and independent variable given in the dataset. If there is more than one independent variable involved, it becomes the Multiple Linear Regression Model. Multiple linear regression is a method for predicting values that are dependent on more than one independent variable.

**Expert Answer:**

Let’s analyze all the statements individually in order to determine the correct option.

**Option 1:**

Option 1 is correct because in linear regression, the given dataset is modeled using a regression equation. This gives the average line where the majority of data value lies that is stated in the option as** the line that best fits a set of data.**

**Option 2:**

The most important feature of any equation is the slope, which tells how much $Y$ changes for every unit change in $X$ (or vice versa). It can be found by dividing both variables. It gives the rate of change of $Y$ per unit $X$, and that means choice 2 is also correct.

**Option 3:**

Option 3 is incorrect as the relationship between dependent and independent variables does not indicate that $X$ causes $Y$.

**Therefore, the correct options are 1 and 2.**

**Alternative Solution:**

From the given options, options** 1** and **2** are true about regression as the statement of option 1 defines the simple regression whereas option 2 also gives the right information about slope that is given as change in $Y$ with respect to $X$.

**Example:**

Which of the following is true about regression with one predictor variable (often called “simple regression”)?

- The Residual Variance/ Error Variance is the square of the Standard Error of the Estimate.
- The intercept in the regression equation \[ Y = a + bX\] is the value of $Y$ when $X$ is zero.
- After conducting a hypothesis test, the slope of the regression equation is nonzero. You can conclude that your predictor variable, $X$, causes $Y$.

In this question, options 1 and 2 are correct whereas option 3 is incorrect.

**Option 1** states the formula for calculating the Standard Error of Estimate. Therefore, it is correct.

If the value of $X$ is zero in the linear regression equation, then the intercept becomes equal to the value of $Y$, which has been stated in **option 2** therefore it is also correct.