In analyzing CTBS scores for math and reading, which test would determine if there is a linear relationship between the two quantitative scores (degrees of freedom unknown)?

In analyzing CTBS scores for math and reading, which test would determine if there is a linear relationship between the two quantitative scores (degrees of freedom unknown)?

• A. 2-sample t-test
• B. Linear Regression t-test (degrees of freedom unknown)
• C. Chi-square test of independence
• D. ANOVA

Answer: (B)

When you want to know if two quantitative scores are related in a linear way, you model one score as a function of the other and test whether that relationship is present. The way to do this is simple linear regression, and the key question is whether the slope of the line that links the two scores is different from zero. If the slope is zero, there’s no linear relationship; if it’s not zero, there’s a linear association. The test used here is a t-test for the slope, and with n observations you estimate two parameters (the intercept and the slope), so the degrees of freedom for the residuals are n minus 2. This approach directly assesses whether a linear relationship exists between the two quantitative measures. Other tests don’t target this specific question. A 2-sample t-test compares means between two groups defined by a categorical variable. The chi-square test of independence looks at associations between categorical variables. ANOVA compares means across groups defined by a categorical factor. None of these directly test whether two continuous variables relate linearly, which is why the regression-based slope test is the appropriate choice.

When you want to know if two quantitative scores are related in a linear way, you model one score as a function of the other and test whether that relationship is present. The way to do this is simple linear regression, and the key question is whether the slope of the line that links the two scores is different from zero. If the slope is zero, there’s no linear relationship; if it’s not zero, there’s a linear association. The test used here is a t-test for the slope, and with n observations you estimate two parameters (the intercept and the slope), so the degrees of freedom for the residuals are n minus 2. This approach directly assesses whether a linear relationship exists between the two quantitative measures.

Other tests don’t target this specific question. A 2-sample t-test compares means between two groups defined by a categorical variable. The chi-square test of independence looks at associations between categorical variables. ANOVA compares means across groups defined by a categorical factor. None of these directly test whether two continuous variables relate linearly, which is why the regression-based slope test is the appropriate choice.