Question $6$

A researcher has estimated a linear model to study the effect of weekly household income $x_i($in $$100)$ on weekly household expenditure on food $y_i($in $ $).$ Using a sample of size $N=40,$ she found that

widehat$(y_i)={83\mathrm{.}42}_{(43\mathrm{.}41)}+10\mathrm{.}21x_i{?}_{(2\mathrm{.}09)},R^2=0\mathrm{.}384$

and ${\displaystyle \underset{i=1}{\overset{N}{}}}(y_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)){)}^2=500,000$ and the sample mean of $x_i$ is $19\mathrm{.}605.$ Choose the wrong statement regarding the $95\%$ confidence interval $($Cl$)$ of the slope parameter.

Selected Answer:

b$.$ The asymptotic $95\%Cl$ is wider than the asymptotic $99\%Cl.$

Answers:

a$.$ The asymptotic $95\%Cl$ is $10\mathrm{.}21+-1\mathrm{.}962\mathrm{.}09.$

b$.$ The asymptotic $95\%Cl$ is wider than the asymptotic $99\%Cl.$

c$.$ The asymptotic $95\%CI$ may not include the true slope parameter.

d$.$ The asymptotic $95\%Cl$ does not include zero according to the information above.

e$.$ The asymptotic $95\%Cl$ is the set of parameter values that are not rejected by a two$-$tailed test at the $5\%$ level.