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p-value

A p-value is the probability of getting results at least as extreme as what you actually observed, assuming the null hypothesis is true. A small p-value suggests your data is unlikely under the null hypothesis, which may lead you to reject it.

The p-value is a conditional probability: given that the null hypothesis H0H_0 is true, it measures the likelihood of obtaining a test statistic equal to or more extreme than the one calculated from the sample data. It is not the probability that H0H_0 is true or false. The p-value is compared against a predetermined significance level α\alpha (commonly 0.05) to decide whether to reject H0H_0.

Key Formula

If p-value≤α, reject H0\text{If } p\text{-value} \leq \alpha, \text{ reject } H_0
Where:
  • p−valuep-value = the probability of observing results as extreme as the data, assuming the null hypothesis is true
  • αα = the significance level (threshold), often set at 0.05
  • H0H₀ = the null hypothesis

Worked Example

Problem: A company claims its light bulbs last an average of 1000 hours. You test 36 bulbs and find a sample mean of 980 hours, with a known population standard deviation of 60 hours. Using a significance level of α = 0.05, find the p-value for a two-tailed test.
Step 1: State the hypotheses. The null hypothesis is that the mean lifetime is 1000 hours. The alternative is that it is not 1000 hours.
H0:μ=1000Ha:μ≠1000H_0: \mu = 1000 \quad H_a: \mu \neq 1000
Step 2: Calculate the test statistic (z-score) using the sample mean, population mean, standard deviation, and sample size.
z=xˉ−μσ/n=980−100060/36=−2010=−2.0z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} = \frac{980 - 1000}{60 / \sqrt{36}} = \frac{-20}{10} = -2.0
Step 3: Find the probability of getting a z-score at least this extreme. From a standard normal table, the area to the left of z = −2.0 is 0.0228.
P(Z≤−2.0)=0.0228P(Z \leq -2.0) = 0.0228
Step 4: Since this is a two-tailed test, double the one-tail probability to account for both directions.
p-value=2×0.0228=0.0456p\text{-value} = 2 \times 0.0228 = 0.0456
Step 5: Compare the p-value to α. Since 0.0456 < 0.05, we reject the null hypothesis.
0.0456<0.05⇒Reject H00.0456 < 0.05 \Rightarrow \text{Reject } H_0
Answer: The p-value is 0.0456. Since this is less than 0.05, there is sufficient evidence to reject the company's claim that the bulbs last an average of 1000 hours.

Visualization

Why It Matters

P-values are central to hypothesis testing across science, medicine, business, and social research. When a clinical trial tests whether a new drug works, or when an engineer checks whether a manufacturing process has shifted, the p-value helps determine whether the observed results are statistically meaningful or could plausibly be due to random chance. Understanding p-values is essential for reading and evaluating research claims critically.

Common Mistakes

Mistake: Interpreting the p-value as the probability that the null hypothesis is true.
Correction: The p-value assumes the null hypothesis is already true. It tells you how likely your data is under that assumption — not how likely the hypothesis itself is.
Mistake: Believing a large p-value proves the null hypothesis is correct.
Correction: A large p-value means you don't have enough evidence to reject the null hypothesis. That's not the same as proving it true — you simply failed to find strong evidence against it.

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