I am writing out a t-test by hand and I am confused about how to mathematically express the step of calculating / looking up the p$p$-value of the t-statistic. This is what I have so far from the following example:
Suppose I take a random sample of size $n=36$ (assume CLT holds) from a population with mean and standard deviation $\mu=67$ and $\sigma = 7$. Also suppose the mean of the sample is $\bar{x} = 71$. Is there sufficient statistical evidence to prove that the mean of the population is actually greater than 67?
My solution thus far is: $$ H_0: \mu = 67 \\ H_A: \mu > 67 \\ $$
$$t = \frac{71 - 67}{\frac{7}{\sqrt{36}}} = \frac{7}{6} \approx 3.428571$$
I understand that the $p$-value can be calculated in R with 1 - pt(3.428571, df=35) resulting in 0.0007848787, but how would I express this mathematically? My best guess is something like:
$$ P(\bar{X} > t_{\text{df}=35}) \approx 0.0007848787 $$
...but I'm hesitant about this because I'm pretty sure $\bar{X}$ is on a different distribution than the t-statistic.
I suppose another way of asking this question is how to mathematically express pt(3.428571, df=35).
I've searched several tutorials for answers but because they tend to be geared more towards beginners they largely gloss over the mathematical expressions (especially for the p$p$-value calculations).
This question is related in the sense that it addresses how to calculate the $p$-value for left-/right-tailed tests but it doesn't really answer my specific question of how to mathematically express the $p$-value in terms of the t-statistic.
