Question

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (January 2011) study of the relationship between task performance and conscientiousness, Exercise 12.94 (p. 766). Recall that y = task performance score (measured on a 30-point scale) was modeled as a function of x₁ = conscientiousness score (measured on a scale of -3 to +3) and x₂ = {1 if highly complex job, 0 if not} using the complete model

$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2+{\beta }_4{x}_1{x}_2+{\beta }_5{\left(x_1\right)}^2{x}_2$

a. Specify the null hypothesis for testing the overall adequacy of the model.

b. Specify the null hypothesis for testing whether task performance score (y) and conscientiousness score (x₁) are curvilinearly related.

c. Specify the null hypothesis for testing whether the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) depends on job complexity (x₂).

Explain how each of the tests, parts a–c, should be conducted (i.e., give the forms of the test statistic and the reduced model).

Short answer

Answer

a. The null hypothesis to test the overall significance of the model will be to check if all the β parameters are zero or not. Mathematically, H₀:${\beta }_0={\beta }_1={\beta }_2={\beta }_3={\beta }_4={\beta }_5=0.$.

b. The null hypothesis to test whether task performance score (y) and conscientiousness score (x₁) are curvilinearly related will be to check if all the β parameters representing theare zero or not. Mathematically, H₀:${\beta }_2={\beta }_5=0.$.

c. The null hypothesis to test whether the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) depends on job complexity (x₂) will be to check if all the β parameters representing (x₁)²the interaction terms are zero or not. Mathematically, H₀:${\beta }_4={\beta }_5=0.$.

d. In part a, F-test would be conducted to check the overall significance of the model. The reduced model in this part would be$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2+{\beta }_4{x}_1{x}_2+{\beta }_5{\left(x_1\right)}^2{x}_2$ . In part b, t-test would be conducted to check the existence of curvilinear relationship between (y) and (x₁). The reduced form model would be$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2+{\beta }_4{x}_1{x}_2$. In part c, t-test would be conducted to check if interaction between variables exists or not. The reduced form model would berole="math" $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2$.

Step-by-step solution

Hypotheses

The null hypothesis to test the overall significance of the model will be to check if all the β parameters are zero or not

Mathematically, H₀:${\beta }_0={\beta }_1={\beta }_2={\beta }_3={\beta }_4={\beta }_5=0.$.

Theorem

The null hypothesis to test whether task performance score (y) and conscientiousness score (x₁) are curvilinearly related will be to check if all the β parameters representing(x₁)² theare zero or not.

Mathematically, H₀:${\beta }_2={\beta }_5=0.$.

Thesis

The null hypothesis to testwhether the curvilinear relationship between task performance score (y) and conscientiousness score (x₁) depends on job complexity (x₂) will be to check if all the β parameters representing the interaction terms are zero or not.

Mathematically, H₀:${\beta }_4={\beta }_5=0.$.

Hypothesis testing

In part a, F-test would be conducted to check the overall significance of the model.

The reduced model in this part would be$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2+{\beta }_4{x}_1{x}_2+{\beta }_5{\left(x_1\right)}^2{x}_2$.

In part b, t-test would be conducted to check the existence of curvilinear relationship between (y) and (x₁). The reduced form model would be$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2+{\beta }_4{x}_1{x}_2$.

In part c, t-test would be conducted to check if interaction between variables exists or not. The reduced form model would be$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{\left(x_1\right)}^2+{\beta }_3{x}_2$.