• The Core Question: If the Null Hypothesis \(H_0\) is true, how likely is it to see data this extreme?
• The Definition: The P-value is a probability, ranging from 0 to 1, calculated from a statistical test.
• The Goal: To provide a standardized metric for “surprisal” in data.
• Interpretation: A small P-value suggests that the observed data is inconsistent with the null hypothesis.

To calculate the P-value, we transform our data into a test statistic. For a sample mean, we use the Z-score:

\[Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\]

Where: * \(\bar{x}\) = Sample Mean * \(\mu_0\) = Population Mean under \(H_0\) * \(\sigma / \sqrt{n}\) = Standard Error

The P-value represents the probability of observing a \(Z\) value at least as extreme as the one calculated.

Where: * \(\bar{x}\) = Sample Mean * \(\mu_0\) = Population Mean under \(H_0\) * \(\sigma / \sqrt{n}\) = Standard Error

The P-value is the area in the tails of the distribution corresponding to this \(z\) value.

groupA <- c(172, 175, 169, 181, 174)
groupB <- c(185, 188, 182, 179, 191)


testResults <- t.test(groupA, groupB)


print(paste("The calculated p-value is:", round(testResults$p.value, 4)))

## [1] "The calculated p-value is: 0.0059"

“We reject \(H_0\) when the observed statistic falls beyond the critical value \(z_{\alpha}\).”

• Probability, not Certainty: The P-value tells us how rare our data is, not if the theory is 100% “true.”
• Thresholds: The \(\alpha = 0.05\) limit is a helpful tool, but context always matters in research.
• Sample Size: As we saw in our maps, more data makes it much easier to achieve significance.

Thank you for your attention!