Calculate p value with statistical precision, sample distributions, and inference data.
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Z = (X̄ - μ₀) / (σ / √n), p = 2 · [1 - Φ(|z|)] (two-tailed)Z / tX̄μ₀σ / sΦ(z)“Testing whether a manufacturing batch differs from standard specifications, producing a test statistic z = 2.15 in a two-tailed test.”
Find the area under the standard normal curve to the left of z = 2.15.
Determine the probability of obtaining a value at or above 2.15 in the positive tail.
Account for extreme deviations in both positive and negative directions.
Because p is less than 0.05, the result is statistically significant.
There is only a 3.16% probability of observing a result this extreme by random chance alone under the null hypothesis. We reject H₀ in favor of the alternative hypothesis.
| P-Value Range | Decision at α = 0.05 | Strength of Evidence | Common Annotation |
|---|---|---|---|
| p < 0.001 | Reject H₀ | Overwhelming evidence against H₀ | *** (Extremely significant) |
| 0.001 ≤ p < 0.01 | Reject H₀ | Strong evidence against H₀ | ** (Highly significant) |
| 0.01 ≤ p < 0.05 | Reject H₀ | Moderate evidence against H₀ | * (Statistically significant) |
| 0.05 ≤ p < 0.10 | Fail to reject H₀ | Weak / trending evidence | ns / trend |
| p ≥ 0.10 | Fail to reject H₀ | No statistically discernible evidence | ns (Not significant) |
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No. The p-value only quantifies statistical significance (the likelihood of observing the data by chance), not practical importance or effect magnitude. With a massive sample size, even trivial differences can produce tiny p-values.
Use a two-tailed test when you want to detect differences in either direction (greater than or less than). Use a one-tailed test only when a directional outcome in the opposite direction is theoretically impossible or irrelevant.
A Z-test is used when the population standard deviation (σ) is known or sample size is very large (n > 30). A Student's t-test is required when σ is unknown and estimated from the sample variance, using degrees of freedom (df = n - 1).
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