P-Value Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of P-Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The p-value is the probability of observing results at least as extreme as the actual data, calculated under the assumption that the null hypothesis is true. A small p-value (typically below 0.05) suggests the observed data is unlikely under the null, providing evidence against it.

P-value answers: 'If nothing special is really happening, how surprising is my data?' A tiny p-value (like 0.01) means your results would be very rare if the null were true - so maybe the null is wrong. A large p-value means your results aren't surprising under the null.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: P-Value uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to p-value but skip the recognition step: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?

Worked Examples

Example 1

medium
For a one-sided upper test with z=1.5z = 1.5, the p-value is approximately P(Z>1.5)≈0.067P(Z > 1.5) \approx 0.067. At α=0.05\alpha = 0.05, what is the decision?

Answer

Fail to reject H0H_0

First step

1
p-value =0.067= 0.067 and α=0.05\alpha = 0.05.

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Example 2

medium
A coin is flipped 2020 times under H0:p=0.5H_0: p = 0.5, and you see 1515 heads. The two-sided p-value is approximately what (binomial tail, accept ≈0.04\approx 0.04)?

Example 3

medium
For a one-sided test with z=−2.33z = -2.33 (lower tail), the p-value is approximately what (use P(Z<−2.33)≈0.01P(Z < -2.33) \approx 0.01)?

Example 4

hard
A study reports p =0.04= 0.04. Without more info, what does NOT this tell us about the effect size?

Example 5

hard
Five tests at family-wise α=0.05\alpha = 0.05 via Bonferroni: each test uses α=\alpha = what?

Example 6

challenge
An exact one-sided permutation p-value uses the proportion of permutations with statistic ≥tobs\ge t_\text{obs}. If 2020 of 10001000 permutations meet this, the p-value is ____.

Example 7

medium
A two-sided z-test has observed z=1.5z=1.5. Upper-tail beyond 1.5 is ≈0.0668\approx 0.0668. Find the two-sided p-value.

Example 8

medium
Suppose under H0H_0 the test statistic TT has CDF FF. The observed value is tobs=2t_\text{obs}=2 with F(2)=0.9772F(2)=0.9772. What is the one-sided (upper) p-value?

Example 9

hard
A t-statistic is t=2.5t=2.5 on 20 df. The upper-tail area beyond 2.5 is about 0.011. Give the two-sided p-value.

Example 10

hard
Compute the two-sided p-value when z=−1.96z = -1.96.

Example 11

challenge
Under H0H_0 a continuous test statistic has p-value distribution Uniform(0,1)(0,1). Find P(p≤0.01∣H0)P(p \le 0.01 \mid H_0).

Example 12

hard
A two-tailed z-test gives z=−2.65z = -2.65. The p-value is approximately 0.008. If α=0.05\alpha = 0.05, should we reject H0H_0?

Example 13

hard
A test gives a p-value of 0.12. Interpret this and state the decision at α=0.05\alpha = 0.05.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Under what assumption is a p-value computed?

Example 2

easy
A small p-value provides evidence ____ the null hypothesis.

Example 3

easy
What threshold is most commonly used to call a p-value 'small'?

Example 4

easy
True or false: the p-value is the probability that the null hypothesis is true.

Example 5

easy
A p-value of 0.01 means the observed (or more extreme) data would occur how often if H0H_0 were true?

Example 6

easy
If p =0.30=0.30 at α=0.05\alpha=0.05, do we reject H0H_0?

Example 7

easy
A larger test statistic (further from 0) generally produces a ____ p-value.

Example 8

easy
Fill in: the p-value measures how ____ the data are, assuming the null is true.

Example 9

medium
A two-sided z-test gives z=2.5z=2.5. The upper-tail area beyond 2.5 is about 0.006. What is the two-sided p-value?

Example 10

medium
Two studies report p =0.049=0.049 and p =0.051=0.051 at α=0.05\alpha=0.05. How different is the actual evidence?

Example 11

medium
A p-value is 0.20. A student concludes 'there is a 20% chance the null is true.' What is the error?

Example 12

medium
A test of a new teaching method gives p =0.002=0.002 but the average score gain is only 0.3 points. What should you report alongside the p-value?

Example 13

medium
If H0H_0 is actually true, what is the distribution of the p-value across many repeated experiments?

Example 14

medium
A one-sided test has z=1.0z=1.0, with upper-tail area about 0.16. Is this evidence against H0H_0 at α=0.05\alpha=0.05?

Example 15

medium
Why does running many tests and reporting only the smallest p-value distort its meaning?

Example 16

medium
A one-sided test of Ha:μ>0H_a:\mu>0 gives z=2.33z=2.33, with upper-tail area about 0.01. At α=0.05\alpha=0.05, decide.

Example 17

medium
A two-sided z-test gives z=1.5z=1.5, with upper-tail area about 0.067. Compute the two-sided p-value.

Example 18

challenge
A two-sided z-test gives z=3z=3. Using that the area beyond ∣z∣=3|z|=3 is about 0.0027 total, state the p-value and the decision at α=0.05\alpha=0.05.

Example 19

challenge
Experiment A: p =0.04=0.04, n=2,000,000n=2{,}000{,}000. Experiment B: p =0.04=0.04, n=20n=20. Why might A's significant result be less impressive than B's?

Example 20

challenge
A researcher computes p =0.03=0.03 and writes 'there is a 97% chance our treatment works.' Identify and correct the two conceptual errors.

Example 21

easy
Under H0H_0, the p-value is the probability of observing data ____ or more extreme than what was seen.

Example 22

easy
At α=0.05\alpha = 0.05, a p-value of 0.020.02 leads to what decision?

Example 23

easy
At α=0.05\alpha = 0.05, a p-value of 0.200.20 leads to what decision?

Example 24

medium
For a two-sided test the p-value from z=1.5z = 1.5 is approximately what?

Example 25

medium
A larger sample size, with the same true effect, tends to produce a ____ p-value.

Example 26

medium
A small p-value gives evidence ____ the null hypothesis.

Example 27

medium
A two-sided p-value can be computed from a one-sided p-value by ____.

Example 28

hard
Under H0H_0, the distribution of the p-value is approximately ____.

Example 29

hard
A study tests 4040 outcomes at α=0.05\alpha = 0.05 but all H0H_0 are true. Expected number of 'significant' results is what?

Example 30

hard
A 'p-hacked' result is one where the researcher ____.

Example 31

medium
For a chi-square test with χ2=10.5\chi^2 = 10.5 and 33 df (critical χ0.052=7.815\chi^2_{0.05} = 7.815), what is the decision at α=0.05\alpha = 0.05?

Example 32

medium
Two studies of the same effect report p =0.04= 0.04 and p =0.06= 0.06. Are their conclusions necessarily different at α=0.05\alpha = 0.05?

Example 33

challenge
A meta-analysis combines kk independent p-values via Fisher's method using −2∑ln⁡(pi)-2 \sum \ln(p_i). Under H0H_0, this statistic follows what distribution?

Example 34

easy
Is a p-value of 0.6 strong evidence against H0H_0?

Example 35

easy
A two-sided z-test gives ∣z∣=1.96|z|=1.96. What is the p-value approximately?

Example 36

easy
At α=0.05\alpha=0.05, the decision when p=0.04p=0.04 is:

Example 37

easy
A one-sided test gives upper-tail area 0.020.02 at the observed zz. What is the one-sided p-value?

Example 38

medium
A p-value of 0.04 is reported. A student says 'the null has 4% probability of being true.' What's wrong?

Example 39

medium
Two studies report p=0.04p=0.04 and p=0.06p=0.06, α=0.05\alpha=0.05. Are these results practically very different?

Example 40

medium
Why do statisticians prefer to report the exact p-value (e.g., 0.012) instead of just 'p < 0.05'?

Example 41

medium
A study has p=0.001p=0.001 for a tiny effect (Δ=0.1\Delta = 0.1 on a 100-point scale) with n=10000n=10000. Is the effect important?

Example 42

medium
A study reports p<0.0001p < 0.0001. Why might you still want to see the test statistic value?

Example 43

hard
Researchers run 100 independent tests. Bonferroni correction uses α/m\alpha/m as the per-test threshold. If overall α=0.05\alpha=0.05, what threshold should each test use?

Example 44

hard
An experiment yields p=0.049p=0.049. A press release says 'the study proved the treatment works.' Why is this wrong?

Example 45

hard
Two independent studies report p=0.04p=0.04 for the same hypothesis. Should we combine them by multiplying p-values?

Example 46

medium
True or false: a p-value above α\alpha means H0H_0 is true.

Example 47

hard
A scientist runs the same test multiple times, choosing the smallest p-value to report. Why does this inflate Type I error?

Example 48

medium
An A/B test reports p =0.20=0.20 after 1000 users. What does this tell us about whether B is better than A?

Example 49

hard
A researcher obtains p=0.03p = 0.03. Would the result be significant at α=0.05\alpha = 0.05? At α=0.01\alpha = 0.01?

Example 50

hard
A hypothesis test gives a p-value of 0.20. What decision would you make at α=0.10\alpha = 0.10 and at α=0.05\alpha = 0.05?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

hypothesis testingprobability basicsampling distribution