Consider you have the suspicion that $Y$ is influenced by two attributes $X_1$ and $X_2$: $$ Y=\theta_0+\theta_1X_1+\theta_2X_2+\theta_3 X_1X_2+U $$ The following data are given. Test on a level of $\alpha=0.05$ if $X_1$ has an influence on $Y$. $$ \begin{array}{c|ccccccccc}\\ \hline x_1 & 4 & 6 & 3 & 8 & 3 & 4 & 9 & 7 & 13 & 10\\ x_2 & 148 & 155 & 138 & 142 & 177 & 159 & 158 & 143 & 153 & 162\\ y & 296 & 303 & 285 & 288 & 324 & 304 & 305 & 291 & 300 & 310\\ \hline \end{array} $$ $$ \begin{array}{c|ccccccccc}\\ \hline x_1 & 5 & 8 & 3 & 6 & 7 & 11 & 6 & 9 & 7 & 10\\ x_2 & 158 & 139 & 153 & 144 & 157 & 167 & 139 & 142 & 155 & 144\\ y & 307 & 287 & 300 & 292 & 305 & 315 & 286 & 187 & 304 & 290\\ \hline \end{array} $$
I do not know how to test this, because of the mixted term $\theta_3 X_1X_2$. If this term would not be there, the testhypothesis would be $$ H_0: \theta_1=0\text{ vs. }H_1: \theta_1\neq 0. $$ But because of the mixted term I do not know how to test.
Update
My only idea is that the test problem might be formulated as the following:
$H_0: \theta_1=\theta_3=0$ vs. $H_1: \theta_i\neq 0$ for at least one $i=1,3$
But I do not know if I have to test this as ONE test (which test statistic?) or as TWO separated test problems (which level each?)
$H_0^1: \theta_1=0$ vs. $H_1^1: \theta_1\neq 0$ (test 1)
$H_0^2: \theta_3=0$ vs. $H_1^2: \theta_3\neq 0$ (test 2)
and then to refuse $H_0$ if at least one of both tests shows that $H_0^i, i=1,2$ has to be refused.
As you already wrote, the test problem is $$ H_0: \theta_1=\theta_3=0\text{ vs. }H_1: \theta_i\neq 0\text{ for at least one }i=1,3. $$ But you should not test this in two separated t-tests! Instead you can write this test problem with a suitable restriction matrix $R$, getting $$ H_0: R\theta=0\text{ vs. }H_1: R\theta\neq 0 $$ and use a $F$-test.