Q:

Construct a $95\%$ confidence interval for $5\beta_4$

Construct a $95\%$ confidence interval for $5\beta_4$.
If this question were about $\beta_4$ without the $5$, I would absolutely know what to do. But I have to idea how the $5$ comes into play. I can't find any examples in my lecture slides or the textbook that have an numerical coefficient before the regression coefficient.
Would I construct the $95\%$ CI for $\beta_4$ and then multiply the upper and lower bounds by $5$? Or is this some clever way of asking for the $95\%$ prediction interval when the $x$ associated with $\beta_4$ is equal to $5$?
Thanks in advance for any clarification you can give!

A:

Confidence interval of $\beta_4$ is constructed using the the following sampling distribution. (My formula here is generic because I don't know how many coefficients you have).
$$\hat{\beta}_4 \sim N( \beta_4, SE^2)\,,$$
where $SE$ is the standard error for $\hat{\beta}_4$. Now let's multiple by 5.
$$5\hat{\beta}_4 \sim N( 5\beta_4, (5\times SE)^2)\,.$$
So the standard error gets multiplied by 5. Now you can make the 95% confidence interval using this distribution.
$$5\hat{\beta}_4 \pm t_{1-\alpha, df} (5 \times SE) = 5\left( \hat{\beta}_4 \pm t_{1-\alpha, df} SE \right)\,.$$
Thus, the upper and lower get multiplied by 5.

