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Question

# Refer to the Development Psychology (Mar. 2003) study of the behavior of elementary school children. The researchers used a quadratic equation to model the level (y) of aggressive fantasies experienced by a child as. a function of the child's age (x). [Note: Level y was measured as an average of responses to six questions (e.g., "Do you sometimes have daydreams about hitting or hurting someone you don't like?"). Responses were measured on a scale ranging from 1 (never) to 5 (always).] a. Write the equation of the hypothesized model for E(y). b. Research psychologists theorized that a child's aggressive fantasies increase with age but at a slower rate of acceleration in older children. Sketch the curve hypothesized by the researchers. c. Set up $H_0$ and $H_a$ for testing the researchers' theory. d. The model was fitted to data collected for over 11,000 elementary school children, with the following results: $\hat{y}=1.926+.097 x-.003 x^{2}$, standard error of $\hat{\beta}_{2}=.001$. Compute the test statistic for the test you set up in part c. e. Use the result you found in part d to make the appropriate conclusion. Take $\alpha=.05$.

Solution

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Given:

\begin{align*} y&=\text{Level of aggressive fantasies} \\ x&=\text{Age} \\ \alpha&=\text{Significance level}=0.05 \end{align*}

(a) The hypothesized model for $E(y)$ is a quadratic model relating $E(y)$ to $x$.

This then implies that the model for $E(y)$ contains the linear term $x$ and the quadratic term $x^2$.

$E(y)=\beta_0+\beta_1x+\beta_2x^2$

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