A researcher studies the relationship between SAT scores, the test-taker's family income, and his or her grade point average (GPA). Data are collected from 24 students. He estimates the following model: SAT = β₀ + β₁ GPA + β₂ Income + ε. The following table summarizes a portion of the regression results. Which of the following is the correct hypotheses for a test about joint significance? A) H₀: β₀ = β₂ = 0; H_A: At least β₀ ≠ 0 B) H₀: β₁ = β₂ = 0; H_A: At least one β_j ≠ 0 C) H₀: β₀ = β₁ = β₂ = 0; H_A: All β_j ≠ 0 D) H₀: β₁ = β₂ = 0; H_A: Both β_j ≠ 0

Question

Quizplus · Accepted Answer

The correct hypotheses for a test about joint significance are those that test whether at least one of the coefficients of the independent variables is different from zero. In this case, the model includes three independent variables (SAT, GPA, and income) with corresponding coefficients (β₀, β₁, and β₂). The null hypothesis, H₀: β₁ = β₂ = 0, tests whether both coefficients for GPA and income are equal to zero. If this hypothesis is rejected, it means that at least one of them is statistically significant in explaining the variation in SAT scores. The alternative hypothesis, H_A: At least one β_j ≠ 0, tests whether at least one of the coefficients for the independent variables is different from zero, indicating that the variables are jointly significant in explaining the variation in SAT scores. Therefore, the correct choice is B.

A Researcher Studies the Relationship Between SAT Scores, the Test-Taker's