Null Hypothesis for Linear Regression

Linear regression is a technique we can use to understand the relationship between one or more predictor variables and a response variable.

Simple Linear Regression

If we only have one predictor variable and one response variable, we can use simple linear regression, which uses the following formula to estimate the relationship between the variables:

\(\large \hat{y} = \beta_0+ \beta_1 x\)

• where:
  \(\large \hat{y}\) : The estimated response value.
  \(\large \beta_0\) : The average value of \(\large y\) when \(\large x\) is \(\large zero\).
  \(\large \beta_1\) : The average change in \(\large y\) associated with a one unit increase in \(\large x\).
  \(\large x\) : The value of the predictor variable.

• \(\large Simple\ linear\ regression\) uses the following \(\large null\ and\ alternative\ hypotheses}\):
  \(\large H0: \beta_1 = 0\)
  \(\large HA: \beta_1 \neq 0\)

• The $\large null\ hypotheses} states that the coefficient \(\large \beta_1\) is equal to zero.
  In other words, there is no statistically significant relationship between the predictor variable, \(\large x\), and the response variable, \(\large y\).

• The $\large alternative\ hypotheses} states that the coefficient \(\large \beta_1\) is not equal to zero. In other words, there is a statistically significant relationship between the predictor variable, \(\large x\), and the response variable, \(\large y\).

Multiple Linear Regression

If we have multiple predictor variables and one response variable, we can use multiple linear regression, which uses the following formula to estimate the relationship between the variables:

\(\large \hat{y} = \beta_0+ \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k\)

• where:
  \(\large \hat{y}\) : The estimated response value.
  \(\large \beta_0\) : The average value of \(\large y\) when all predictor variables are equal to \(\large zero\).
  \(\large \beta_i\) : The average change in \(\large y\) associated with a one unit increase in \(\large x_i\).
  $\large x_i $ : The value of the predictor variable $\large x_i $.

• \(\large Multiple\ linear\ regression\) uses the following \(\large null\ and\ alternative\ hypotheses}\):
  \(\large H0: \beta_1 = \beta_2 = \cdots = \beta_k = 0\)
  \(\large HA: \beta_1 = \beta_2 = \cdots = \beta_k \neq 0\)

• The $\large null\ hypotheses} states that all coefficients in the model are equal to zero.
  In other words, none of the predictor variables $\large x_i $ have a statistically significant relationship with the response variable, \(\large y\).

• The $\large alternative\ hypotheses} states that not every coefficient is \(\large simultaneously\) equal to zero.

The following examples show how to decide to reject or fail to reject the null hypothesis in both simple linear regression and multiple linear regression models.

Example 1: Simple Linear Regression

Suppose a professor would like to use the number of hours studied to predict the exam score that students will receive in his class. He collects data for 20 students and fits a simple linear regression model.

The following screenshot shows the output of the regression model:

Output of simple linear regression in Excel

The fitted simple linear regression model is:

Exam Score = 67.1617 + 5.2503*(hours studied)

To determine if there is a statistically significant relationship between hours studied and exam score, we need to analyze the overall F value of the model and the corresponding p-value:

Overall F-Value: 47.9952
P-value: 0.000
Since this p-value is less than .05, we can reject the null hypothesis. In other words, there is a statistically significant relationship between hours studied and exam score received.

Example 2: Multiple Linear Regression

Suppose a professor would like to use the number of hours studied and the number of prep exams taken to predict the exam score that students will receive in his class. He collects data for 20 students and fits a multiple linear regression model.

The following screenshot shows the output of the regression model:

Multiple linear regression output in Excel

The fitted multiple linear regression model is:

Exam Score = 67.67 + 5.56(hours studied) – 0.60(prep exams taken)

To determine if there is a jointly statistically significant relationship between the two predictor variables and the response variable, we need to analyze the overall F value of the model and the corresponding p-value:

Overall F-Value: 23.46
P-value: 0.00
Since this p-value is less than .05, we can reject the null hypothesis. In other words, hours studied and prep exams taken have a jointly statistically significant relationship with exam score.

Note: Although the p-value for prep exams taken (p = 0.52) is not significant, prep exams combined with hours studied has a significant relationship with exam score.

Additional Resources

Understanding the F-Test of Overall Significance in Regression
How to Read and Interpret a Regression Table
How to Report Regression Results
How to Perform Simple Linear Regression in Excel
How to Perform Multiple Linear Regression in Excel