Question

Assertiveness and leadership. Management professors at Columbia University examined the relationship between assertiveness and leadership (Journal of Personality and Social Psychology, February 2007). The sample represented 388 people enrolled in a full-time MBA program. Based on answers to a questionnaire, the researchers measured two variables for each subject: assertiveness score (x) and leadership ability score (y). A quadratic regression model was fit to the data, with the following results:

a. Conduct a test of overall model utility. Use$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$ .

b. The researchers hypothesized that leadership ability increases at a decreasing rate with assertiveness. Set up the null and alternative hypotheses to test this theory.

1. Use the reported results to conduct the test, part b. Give your conclusion(at$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$ )in the words of the problem.

Short answer

a) At 95% significance level, ${\mathbf{\beta }}_{\mathbf{1}}\ne {\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$

b) The null hypothesis is whether ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$while the alternate checks if the value of ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{<}\mathbf{0}$Mathematically, while${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{<}\mathbf{0}$

c) Therefore,${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$,This means that the parabola doesn’t have a curvature and it essentially is a straight line.

Step-by-step solution

Overall goodness of fit

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

${\mathbf{H}}_{\mathbf{a}}\mathbf{:}$At least one of the parameters ${\mathbf{\beta }}_{\mathbf{1}}$or ${\mathbf{\beta }}_{\mathbf{2}}$is non zero

Here, F test statistic $\mathbf{=}\frac{\mathbf{SSE}}{\mathbf{n}\mathbf{-}\mathbf{(}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}}$

${\mathbf{H}}_{\mathbf{0}}$is rejected if p-value < $\mathbf{\alpha }$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since for x and ${\mathbf{x}}^{\mathbf{2}}\mathbf{p}$ - value is less than 0.01

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

Thus,${\mathbf{\beta }}_{\mathbf{1}}\ne {\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$.

Significance of  β2

The null hypothesis is whether ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$while the alternate checks if value of $\mathbf{t} \mathbf{<}{\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{199}}$

Mathematically, ${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{=}\mathbf{0}$while${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{<}\mathbf{0}$

Consequence of  β2

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{<}\mathbf{0}$

Here, t-test statistic$\mathbf{=}\frac{\stackrel{\mathbf{⏜}}{\mathbf{\beta }}}{{\mathbf{s\beta }}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{088}}{\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{97}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0221}$

Value of ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{385}}$ is 1.645

${\mathbf{H}}_{\mathbf{0}}$ is rejected if t statistic > ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{385}}$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since $\mathbf{t} \mathbf{<}{\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{199}}$

Not sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$ at 95% confidence interval.

Hence,${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$.

This means that the parabola doesn’t have a curvature and it essentially is a straight line.