# 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!

• Might it mean "parameter B4 from Equation 5"? – James Phillips Oct 27 '18 at 15:56

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. $$5\hat{\beta}_4 \sim N( 5\beta_4, (5\times SE)^2)\,.$$
$$5\hat{\beta}_4 \pm t_{1-\alpha, df} (5 \times SE) = 5\left( \hat{\beta}_4 \pm t_{1-\alpha, df} SE \right)\,.$$