P-value for t-score

Question

With a z-score you can look-up the p-value (percentage) in a table. For t-scores you just get a table to look up the t-score for your desired alpha level given the degrees of freedom you have. You can compare that score with your t-score but what if you wanted a p-value instead?

This is where statistical software like R, Python, SAS, Stata, and SPSS come in (I prefer the first two). Excel might even have some tools like “pt” in R. But I think this gets at how the reject/not methodology came to be 100 years ago when they had to make decisions based on tables like you have and couldn’t calculate the p-value on a computer. They either landed above or below the specified threshold. — Dave, Feb 11 '21 at 05:17
@Dave Your history seems a little skewed. The standard tools for many centuries (at least four) were tables and interpolation techniques. That was the basis for computing t-statistic p-values until computers started becoming more accessible to researchers (in the 1960's) and everyone else (in the late 1980's). A great many "numerical quadrature" techniques were also in vogue and used by engineers and mathematicians working with pencil and paper (or adding machines). The cumulative Student t distributions are particularly amenable to numerical integration. — whuber, Feb 11 '21 at 19:33

score 2 · Accepted Answer · answered Feb 11 '21 at 11:45

In short: ask a machine.

Say your t-statistic is 2.14 on 39 degrees of freedom.

The table probably gives you t for conventional values of alpha (here, three of them). They're called 'p' in R's quantile function for the t-distribution, qt

alphas <- c(0.001, 0.01, 0.05)

qt(p = alphas, df = 39, lower.tail = FALSE)
[1] 3.312788 2.425841 1.684875

whereas you want 'p' from a particular value of t. You can get it from R's t-distribution CDF function, pt.

pt(2.14, df = 39, lower.tail = FALSE)
[1] 0.01933356

Set 'p' to 0.01933356 in qt to recover 2.14

P-value for t-score

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