statistic $t$-test for $2$ means I'm just doing the exercises from Stock&Watson "Introduction to econometrics". $3.12$ is about comparing men and women salaries
mean for men: $8200,$ SD $= 450, n_1=120$
mean for women: $7900,$ SD $=520, n_2=150$
I took $t= \dfrac{X_1-X_2}{\frac{\text{spooled}}{0.5(n_1+n_2)}}$
I obtained $t=5.03$
I know I have to reject H$0.$ They're asking me to calculate $p$-value. I looked for it in the tables. There is till $2.9.$ How to calculate $p$-value in that case? They're asking if the company is guilty of gender discrimination in its compensation policies. If I rejected H$0$ should I claim they are guilty?
 A: The number of degrees of freedom is so high that your $t$-test is effectively a $z$-test.  You can get an approximate $p$-value for a two-sided hypothesis by simply looking up your test statistic in a $z$-table.
Note that I chose to employ the Welch $t$-test rather than the two-sample $t$-test with the assumption of equal variances as you seem to have done.*  Thus, the value of my test statistic is $$T \approx 5.07806, \quad df \approx 509.038,$$ which if we use the $z$-table, gives $$p \approx 3.813 \times 10^{-7},$$ whereas using the exact $t$ distribution with $df = 509.038$, would give instead $$p \approx 5.35858 \times 10^{-7}.$$  (Again, all $p$-values are calculated for a two-sided test.)
*Note.  I should also point out that your computation of the test statistic, even for an independent two-sample $t$-test with equal variances, appears to be incorrect.  I have $$T = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \approx 4.99739.$$  Your formula does not specify how you compute the pooled standard deviation, and I cannot replicate your result based on the formula you stated.
