A p-value (probability value) quantifies the strength of evidence in statistical hypothesis testing, helping researchers evaluate whether their findings are statistically significant. In this framework, the null hypothesis (H₀) serves as the default assumption, positing no effect, difference, or relationship (e.g., “no association between variables”). The p-value represents the probability of observing results as extreme as—or more extreme than—those measured in the study, assuming the null hypothesis is true. In simpler terms, it answers: If the null hypothesis were correct, how unusual are the observed data?

Interpreting p-Values

• A smaller p-value suggests stronger evidence against the null hypothesis. For instance, p = 0.01 implies a 1% chance of observing such extreme results under H₀, making it more compelling to reject H₀ than p = 0.10.
• A common threshold for significance is p ≤ 0.05. If the p-value falls at or below this level, the result is deemed statistically significant, and H₀ is rejected in favor of the alternative hypothesis (H₁). Conversely, p > 0.05 indicates insufficient evidence to discard H₀.

Using Z-Scores for p-Values (Normal Distribution)
When data follows a normal distribution, a Z-score table (standard normal table) can calculate p-values. Here’s how:

1. Calculate the Z-Score:
   This standardizes the observed data relative to the null hypothesis. For a sample mean:
   [ Z = \frac{\bar{X} – \mu}{\sigma / \sqrt{n}} ]
   Where:
   • (\bar{X}) = sample mean
   • (\mu) = population mean under H₀
   • (\sigma) = population standard deviation
   • (n) = sample size
2. Consult the Z-Table:
   The table provides the cumulative probability (area under the curve) to the left of the Z-score. For example, a Z-score of 1.96 corresponds to a cumulative probability of 0.9750.
3. Determine the p-Value:
   • One-tailed test (e.g., testing if a parameter is greater than a value):
     [ p\text{-value} = 1 – \text{Cumulative Probability} ]
     For Z = 1.96: (p = 1 – 0.9750 = 0.025).
   • Two-tailed test (e.g., testing for any difference, positive or negative):
     [ p\text{-value} = 2 \times (1 – \text{Cumulative Probability}) ]
     For Z = 1.96: (p = 2 \times 0.025 = 0.05).
4. Compare to Significance Level ((\alpha)):
   • If (p \leq \alpha) (e.g., 0.05), reject H₀.
   • If (p > \alpha), retain H₀.

Example:
A two-tailed test yields Z = 2.0. The cumulative probability for Z = 2.0 is 0.9772.
[ p\text{-value} = 2 \times (1 – 0.9772) = 0.0456 ]
Since 0.0456 < 0.05, H₀ is rejected.

Key Considerations:

• Test Type: Choose one-tailed or two-tailed based on the research question.
• Z-Table Direction: Confirm whether your table provides area to the left or right of Z.
• Context Matters: Statistical significance does not equate to practical importance. Always interpret results alongside real-world relevance.

This approach bridges theoretical statistics with practical application, enabling researchers to make data-driven decisions using the properties of the normal distribution.