Question

When we assess a linear regression model, there are several tests we can use. What is the test called that determines whether the predictor variables x1, x2,..., xk have a joint statistical influence on y?

A. Test for nonzero slope coefficient
B. Test of joint significance
C. Test for goodness of fit
D. Test of individual significance

Answer 1

The F-test of overall significance is used in linear regression models to determine whether multiple predictors have a joint statistical influence on the dependent variable. It contrasts a model with predictors to one without, checking if the relationship is more than coincidental.

Step-by-step explanation:

The test that determines whether the predictor variables x1, x2, ..., xk have a joint statistical influence on the dependent variable y in a linear regression model is known as the F-test of overall significance. This test is used to check if there is a linear relationship between the predictor variables and the dependent variable beyond what you would expect by chance. It is based on the premise that the linear model estimates the relationship drawn from a sample and reflects the population from which the sample is taken.

Here's an overview of the steps involved in assessing a linear regression model concerning the question:

1. Identify the independent and dependent variables: The predictor variables (x1, x2, ..., xk) are independent, and the outcome variable (y) is dependent.
2. Draw a scatter plot: This visual representation can help indicate the presence or absence of a relationship.
3. Use regression to find the line of best fit and the correlation coefficient: The least-squares method is commonly used to calculate this line, which is expressed as î = a + bx.
4. Interpret the significance of the correlation coefficient: Using hypothesis testing, we can determine if the observed correlation coefficient is significantly different from zero, indicating a relationship.
5. Assess the linear relationship: Consider both the strength and the direction of the relationship, including an evaluation of residuals.

The F-test of overall significance assesses if the model as a whole has predictive capability by comparing it to a model with no predictors. The null hypothesis typically states that the model with no predictors is sufficient, meaning the coefficients of the predictor terms are all zero. A significant F-test indicates that the regression model predicts the dependent variable better than the mean of the dependent variable alone.