Question

Personality traits and job performance. When attempting to predict job performance using personality traits, researchers typically assume that the relationship is linear. A study published in the Journal of Applied Psychology (Jan. 2011) investigated a curvilinear relationship between job task performance and a specific personality trait—conscientiousness. Using data collected for 602 employees of a large public organization, task performance was measured on a 30-point scale (where higher scores indicate better performance) and conscientiousness was measured on a scale of -3 to +3 (where higher scores indicate a higher level of conscientiousness).

a. The coefficient of correlation relating task performance score to conscientiousness score was reported as r = 0.18. Explain why the researchers should not use this statistic to investigate the curvilinear relationship between task performance and conscientiousness.

b. Give the equation of a curvilinear (quadratic) model relating task performance score (y) to conscientiousness score (x).

c. The researchers theorized that task performance increases as level of conscientiousness increases, but at a decreasing rate. Draw a sketch of this relationship.

d. If the theory in part c is supported, what is the expected sign of${\mathbf{\beta }}_{\mathbf{2}}$in the model, part b?

e. The researchers reported${\hat{\mathbf{\beta }}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{32}$with an associated p-value of less than 0.05. Use this information to test the researchers’ theory at$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$

Short answer

a. Researchers cannot use the correlation coefficient to investigate the curvilinear relationship amongst the variable as the correlation coefficient indicates the extent to which two variables move together but does not account for the curvilinear relationship the two variables might have.

b. The quadratic model equation relating task performance (y) to conscientiousness score (x) is $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}$.

c. Graph

d. The curve is downward sloping, the value of β₂ which measures the slope of the curvature will be negative.

e. At 95% confidence level, ${\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$.

Step-by-step solution

Interpretation of r

The coefficient of correlation value, r = 0.18 which indicates a positive relation between y and x. However, researchers cannot use the correlation coefficient to investigate the curvilinear relationship amongst the variable as the correlation coefficient indicates the extent to which two variables move together but does not account for the curvilinear relationship the two variables might have.

Second-order model equation

The quadratic model equation relating task performance (y) to conscientiousness score (x) is $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}$.

Graph

The relationship between y and x where y increases with x but at a decreasing rate can be shown using a downward sloping curve.

Sign of β2

Since the curve is downward sloping, the value of ${\mathbf{\beta }}_{\mathbf{2}}$which measures the slope of the curvature will be negative.

Significance of β2

$$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0} \end{gathered}$$

Here, t-test statistic$\mathbf{=}\frac{{\hat{\mathbf{\beta }}}_{\mathbf{3}}}{\mathbf{s}{\hat{\mathbf{\beta }}}_{\mathbf{3}}}$

${\mathbf{H}}_{\mathbf{0}}$is rejected if p-value < $\mathbf{\alpha }$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, it is mentioned that p-value is less than 0.05

Sufficient evidence to reject${\mathbf{H}}_{\mathbf{0}}$ at 95% confidence interval.

Therefore, ${\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$