P-Values

A p-value is the probability of getting a sample result at least as extreme as the one you observed, assuming the null hypothesis is true. A small p-value means your data would be rare if H₀ were true, so it counts as evidence against H₀. If the p-value is at or below the significance level α, you reject H₀.

Before this lesson: Hypothesis Testing and Normal Probability Calculations

What a p-value measures

Every test starts by assuming H₀ is true. The p-value then asks: in that world, how often would random sampling alone give a result this far from the H₀ value, or farther?

“Extreme” means in the direction of the alternative hypothesis Hₐ. For Hₐ with >, it is the right tail. For <, it is the left tail. For ≠, it is both tails, so you double one tail.

The smaller the p-value, the stronger the evidence against H₀. A p-value is a probability about the data, computed by assuming H₀. It is not the probability that H₀ is true.

How to find and use a p-value

For a t statistic, use tcdf(lower, upper, df) the same way (see t-distribution).

\text{p-value} \le \alpha \Rightarrow \text{reject } H_0 \qquad \text{p-value} \gt \alpha \Rightarrow \text{fail to reject } H_0

How to interpret it: “Assuming H₀ is true, there is a [p-value] probability of getting a sample result as extreme as ours, or more extreme, by chance alone.”

P-value examples

Example 1: A right-tailed p-value

A test of H₀: p = 0.30 against Hₐ: p > 0.30 gives z = 1.5. Find the p-value and make a decision at α = 0.05.

P(Z \ge 1.5) = \text{normalcdf}(1.5, 10^{99})

\approx 0.0668

0.0668 \gt 0.05

Answer: the p-value is about 0.0668, which is greater than 0.05, so fail to reject H₀. The data don’t give convincing evidence that p > 0.30.

Example 2: A two-sided p-value

A test of H₀: p = 0.5 against Hₐ: p ≠ 0.5 gives z = −2.1. Find the p-value. What is the decision at α = 0.05? At α = 0.01?

P(Z \le -2.1) = \text{normalcdf}(-10^{99}, -2.1)

\approx 0.0179

2 \cdot 0.0179 = 0.0358

0.0358 \lt 0.05

0.0358 \gt 0.01

Answer: the p-value is about 0.0358. Reject H₀ at α = 0.05, but fail to reject at α = 0.01. The same data can lead to different decisions, which is why α is chosen before the test.

Example 3: A p-value from a t statistic

A one-sample t-test of Hₐ: μ > μ₀ gives t = 2.4 with df = 14. Find the p-value.

P(t \ge 2.4) = \text{tcdf}(2.4, 10^{99}, 14)

\approx 0.0154

0.0154 \lt 0.05

Answer: the p-value is about 0.0154, so reject H₀ at α = 0.05. A t statistic this large would happen only about 1.5% of the time if H₀ were true.

Example 4: Interpret a p-value (word problem)

A school says 25% of its students bike to school. A student council survey of 200 randomly chosen students finds 62 who bike. Test H₀: p = 0.25 against Hₐ: p > 0.25, and interpret the p-value.

\hat{p} = \dfrac{62}{200} = 0.31

0.25 \cdot 0.75 = 0.1875

\dfrac{0.1875}{200} = 0.0009375

\text{SE} = \sqrt{0.0009375} \approx 0.03062

0.31 - 0.25 = 0.06

z = \dfrac{0.06}{0.03062} \approx 1.960

P(Z \ge 1.960) = \text{normalcdf}(1.960, 10^{99})

\approx 0.025

Answer: the p-value is about 0.025. Interpretation: if 25% of all students at the school bike, there is about a 2.5% chance that a random sample of 200 would show 31% or more bikers. That is rare enough to reject H₀ at α = 0.05.

Example 5: Significant but small

An online store tests whether a new button changes the click rate from p = 0.5. With 40,000 visitors, p̂ = 0.505. Find the two-sided p-value. Does the result matter in practice?

0.5 \cdot 0.5 = 0.25

\dfrac{0.25}{40000} = 0.00000625

\text{SE} = \sqrt{0.00000625} = 0.0025

z = \dfrac{0.505 - 0.5}{0.0025} = \dfrac{0.005}{0.0025}

= 2

P(Z \ge 2) \approx 0.02275

2 \cdot 0.02275 = 0.0455

Answer: the p-value is about 0.0455, so the result is statistically significant at α = 0.05. But the change is only half a percentage point. A huge sample can make a tiny difference significant, so always ask whether the size of the effect matters.

Common mistakes

Related methods

Comparing p-values with confidence intervals

A p-value says whether the data are consistent with one specific value. A confidence interval shows all the plausible values at once (see confidence intervals). Reporting both gives the clearest picture.

Where p-values come from

Each test has its own test statistic and distribution. The next lesson, One-Proportion z-Test, puts every step together for a single proportion.

Practice problems

1. A right-tailed z-test gives z = 1.2. Find the p-value.
2. A left-tailed z-test gives z = −2.5. Find the p-value.
3. A two-sided z-test gives z = 1.8. Find the p-value.
4. A test gives p-value = 0.04. What is the decision at α = 0.05? At α = 0.01?
5. A left-tailed t-test gives t = −1.9 with df = 20. Find the p-value.
6. A test of whether a coin favors heads gives a p-value of 0.12. Interpret it in context.
7. True or false: a p-value of 0.03 means there is a 3% chance that H₀ is true.
8. A test gives p-value = 0.027. For which significance levels would you reject H₀?
9. A two-sided t-test gives t = 2.6 with df = 11. Find the p-value.
10. Which gives stronger evidence against H₀: a p-value of 0.002 or 0.04?

P(Z \ge 1.2) = \text{normalcdf}(1.2, 10^{99})

\approx 0.1151

P(Z \le -2.5) = \text{normalcdf}(-10^{99}, -2.5)

\approx 0.0062

P(Z \ge 1.8) = \text{normalcdf}(1.8, 10^{99})

\approx 0.0359

2 \cdot 0.0359 = 0.0718

0.04 \lt 0.05

0.04 \gt 0.01

P(t \le -1.9) = \text{tcdf}(-10^{99}, -1.9, 20)

\approx 0.036

P(t \ge 2.6) = \text{tcdf}(2.6, 10^{99}, 11)

\approx 0.01235

2 \cdot 0.01235 = 0.0247

0.002 \lt 0.04

P-value FAQ

What does a p-value tell you?

It tells you how likely a result as extreme as yours would be if H₀ were true. A small p-value means the data are hard to explain by chance alone.

What is a good p-value?

There is no single “good” value. A p-value at or below your significance level, usually 0.05, counts as statistically significant. Smaller p-values give stronger evidence against H₀.

How do you find a p-value for a two-tailed test?

Find the tail area beyond your test statistic on one side, then multiply by 2. On a TI-84, use 2 · normalcdf(|z|, 10^99) for a z statistic.

Is a p-value of 0.05 significant?

At α = 0.05, yes: a p-value equal to α leads you to reject H₀. Many researchers also report the exact p-value so readers can judge the strength of the evidence.

Summary

• The p-value is the probability, assuming H₀ is true, of a result at least as extreme as the one observed.
• Use the tail that matches Hₐ; double one tail for a two-sided test.
• If p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.
• Smaller p-values mean stronger evidence against H₀.
• Statistical significance is not the same as practical importance.

Previous lesson: Hypothesis Testing

Next lesson: One-Proportion z-Test

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