The transportation department collected a sample of n=512 individuals who commute to their work place. The variable TRVLi represents the travel time (in minutes) it takes individual i to travel to his/her work (an average for that individual). The sample also includes the variable INCi which is the annual income (in thousands of dollars) that individual i earned (for 2014). The results of a linear regression were as follows: $$ \widehat{TRVL_i} = 13.05_{(4.2)} + 1.42 _{(0.39)} \space INC_i $$ Using a 95% confidence level, test the hypothesis that any additional $1000 dollars increase to annual income increase the travel time by 2 minutes on average.
$H_0: \beta_1 = 1$
$H_A: \beta_1 \neq 1$
t ratio = $\frac{1.42-1}{0.39}$ = 1.076923077
$Z_{0.975} = 1.96$
Since the t ratio < z value we cannot reject the hypothesis. I gave the problem a shot but I'm not sure if I'm correct. I was hoping that someone could verify my work and answer. Thank you!
Denote $Y$ travel time and $X$ annual income, then $$ E[Y|X=x] = \beta_0 + \beta_1x, $$ where under $H_0$ you have that $\frac{\Delta y}{\Delta x}=\Delta y \approx\frac{\partial }{\partial x}E[Y|X=x]=\beta_1=2$. Now you can use the fact that $\hat{\beta}/0.39$ is distributed $\mathcal{T}$ with $n=512-2$ df to construct the CI for the hypothesis test.