How to interpret the value of $t=12.156$ in significance testing?

Question

I am unsure on how to interpret the value of $t_{1}=12.156 $ and similarly for $t_{2} $ as in the tables for the t-distribution for $17$ degrees of freedom there is no value for $t=12.156$? I am not sure how the conclusion has been drawn in the answer. Also from the conclusion it seems a significance test has been used, how would it work if we found the p-value and worked from there.

Any help would be much appreciated!

Answer 1

Notice that for $\mathrm{df}=17$, even at $0.001$ level of significance, the critical value is $t=3.965\ll10$. Hence, this would mean that $$t_1,t_2\gg t_\text{crit}$$ for any reasonable level of significance, leading to rejection of $H_0$. As there is much less than $0.1\%$ chance that $H_0$ is true and still produces such test statistics.

Answer 2

That is your standardized test statistic. It is very big so you don't see it on the table (usually $t>4$ is enough to get it off the table). The bigger your $t$ value, the less likely it is that your data would be collected given the null hypothesis.