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.

Step by step solution

01

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

02

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${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{<}\mathbf{0}$

03

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.

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