Solution

Writing

Y_{i} = medv_{i}

L_{i} = lstat_{i}

, and

A_{i} = age_{i}

Y_{i} = β_{0} + β_{1} L_{i} + β_{2} A_{i} + β_{3} L_{i} A_{i} + ε_{i}, ε_{i} \sim iid N (0, σ^{2}) .

The R formula lstat * age includes both main effects and their interaction term.

Three notable features are:

The fitted model is highly significant overall: its $F$ statistic is $209.3$ on $(3, 502)$ degrees of freedom with $p < 2.2 \times 1 0^{- 16}$ . Its $R^{2} = 0.5557$ , however, means that about $44%$ of the observed response variation remains unexplained, and the residual standard deviation is about $6.15$ house-value units.

At fixed age, the fitted slope with respect to lstat is
$- 1.3921 + 0.004156 A .$
(26)
The large negative main coefficient is strongly significant. The positive interaction is significant at the $5%$ level ( $p = 0.0252$ ), indicating that this negative association becomes slightly weaker for older housing tracts.

The age main-effect estimate is essentially zero with $p = 0.9711$ , but it is the age effect specifically at $L = 0$ and should not be interpreted separately from the significant interaction. The diagnostic plots also show curved residual structure, increasing spread, a heavy upper tail with observations 215, 372, and 373, and substantial influence from observation 215. These features cast doubt on linearity, constant variance, and Gaussian residual assumptions.

Solved by gpt-5.6-sol high.

 Ancestors (10)