线性回归分析-HW5

HW5.
Data.

Prob-1

1-a

Consistent, as the age groups are similar in size.

1-b

The residuals present good normality.

1-c

The two plots provide the same information as $\hat{Y}_{i}$ is a linear transformation of $X_{i}$.

The two plots ascertain $\mathbb{E}(\varepsilon\mid X)=0$ and $\mathbb{V}[\varepsilon\mid X]=\sigma^{2}$.

1-d

q <- qnorm(res)
cor(q$x,q$y)

The coefficient of correlation: 0.9896158

Use the correlation test

$$t=\frac{r}{\sqrt{1-r^{2}}}\sqrt{n-2}\sim t_{n-2}$$

Here $t=52.434,n=60$.

The results are

We have enough evidence to say that they have linear correlation.

1-e

Suppose

$$\sigma_{i}^{2}=\exp(\alpha_{0}+\alpha_{1}X_{i})$$

The alternatives:

$$\begin{align} H_{0} & :\alpha_{1}=0 \\ H_{1} & :\text{o.w.} \end{align}$$

The test statistic:

$$LM=\frac{SSR^{*}}{2}\big/\left( \frac{SSE}{n} \right)^{2}=\frac{31833}{2}\big/\left( \frac{3874.4}{60} \right)^{2}=3.817167$$

The decision rule:

$$LM>\chi^{2}_{0.99,1}=6.634897$$

The conclusion: cannot reject.

Prob-2

It is saying that the median is smaller than the mean, which usually indicates that the distribution is right-skewed.