One tail or two?
Use a two-tailed test when you care about a difference in either direction — the usual default, and the honest choice unless you decided otherwise before seeing the data. Use a one-tailed test only when a difference in the opposite direction would be meaningless to you.
The shaded region on the curve is exactly what the p-value measures. Switching from two-tailed to one-tailed halves the p-value, which is precisely why choosing the tail after looking at your results is not legitimate.
What p actually measures
The p-value is the probability of seeing a result at least this extreme if the null hypothesis were true. A small p means your data would be surprising under the null — that is all. It is not the probability that the null is true, not the probability your finding will replicate, and not a measure of effect size.
A p of 0.049 and a p of 0.051 are essentially the same evidence; the 0.05 line is a convention, not a discovery threshold. Report the interval alongside the p-value and you give the reader something far more useful.
Frequently asked questions
My p-value shows as scientific notation — what does 3.2e-5 mean?
It means 0.000032. Very small p-values are shown that way because the decimal form becomes unreadable, and the exact table below gives the full precision.
Can I rely on this for coursework or research?
Verified against published critical values for one and two-tailed tests. One-tailed and two-tailed give different answers, and which is correct depends on the hypothesis, not the data.
Is my data uploaded?
No. The test statistic never leaves your machine.
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