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.
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=1.5, the p-value is approximately P(Z>1.5)≈0.067. At α=0.05, what is the decision?
Answer
Fail to reject H0
First step
1
p-value =0.067 and α=0.05.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
A coin is flipped 20 times under H0:p=0.5, and you see 15 heads. The two-sided p-value is approximately what (binomial tail, accept ≈0.04)?
Example 3
medium
For a one-sided test with z=−2.33 (lower tail), the p-value is approximately what (use P(Z<−2.33)≈0.01)?
Example 4
hard
A study reports p =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 via Bonferroni: each test uses α= what?
Example 6
challenge
An exact one-sided permutation p-value uses the proportion of permutations with statistic ≥tobs. If 20 of 1000 permutations meet this, the p-value is ____.
Example 7
medium
A two-sided z-test has observed z=1.5. Upper-tail beyond 1.5 is ≈0.0668. Find the two-sided p-value.
Example 8
medium
Suppose under H0 the test statistic T has CDF F. The observed value is tobs=2 with F(2)=0.9772. What is the one-sided (upper) p-value?One-sided (upper) p-value = 1 − F(2) = 1 − 0.9772 = 0.0228
Example 9
hard
A t-statistic is t=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=−1.96.
Example 11
challenge
Under H0 a continuous test statistic has p-value distribution Uniform(0,1). Find P(p≤0.01∣H0).
Example 12
hard
A two-tailed z-test gives z=−2.65. The p-value is approximately 0.008. If α=0.05, should we reject H0?
Example 13
hard
A test gives a p-value of 0.12. Interpret this and state the decision at α=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 H0 were true?
Example 6
easy
If p =0.30 at α=0.05, do we reject H0?
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=2.5. The upper-tail area beyond 2.5 is about 0.006. What is the two-sided p-value?Two-sided p-value: both tails beyond |z|
Example 10
medium
Two studies report p =0.049 and p =0.051 at α=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 but the average score gain is only 0.3 points. What should you report alongside the p-value?
Example 13
medium
If H0 is actually true, what is the distribution of the p-value across many repeated experiments?
Example 14
medium
A one-sided test has z=1.0, with upper-tail area about 0.16. Is this evidence against H0 at α=0.05?One-sided (upper) p-value: tail area beyond z
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>0 gives z=2.33, with upper-tail area about 0.01. At α=0.05, decide.One-sided p-value: tail area beyond z
Example 17
medium
A two-sided z-test gives z=1.5, with upper-tail area about 0.067. Compute the two-sided p-value.Two-sided p-value: both tails beyond |z|
Example 18
challenge
A two-sided z-test gives z=3. Using that the area beyond ∣z∣=3 is about 0.0027 total, state the p-value and the decision at α=0.05.Two-sided p-value: both tails beyond |z|
Example 19
challenge
Experiment A: p =0.04, n=2,000,000. Experiment B: p =0.04, n=20. Why might A's significant result be less impressive than B's?
Example 20
challenge
A researcher computes p =0.03 and writes 'there is a 97% chance our treatment works.' Identify and correct the two conceptual errors.
Example 21
easy
Under H0, the p-value is the probability of observing data ____ or more extreme than what was seen.
Example 22
easy
At α=0.05, a p-value of 0.02 leads to what decision?
Example 23
easy
At α=0.05, a p-value of 0.20 leads to what decision?
Example 24
medium
For a two-sided test the p-value from z=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 H0, the distribution of the p-value is approximately ____.
Example 29
hard
A study tests 40 outcomes at α=0.05 but all H0 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 and 3 df (critical χ0.052=7.815), what is the decision at α=0.05?
Example 32
medium
Two studies of the same effect report p =0.04 and p =0.06. Are their conclusions necessarily different at α=0.05?
Example 33
challenge
A meta-analysis combines k independent p-values via Fisher's method using −2∑ln(pi). Under H0, this statistic follows what distribution?
Example 34
easy
Is a p-value of 0.6 strong evidence against H0?
Example 35
easy
A two-sided z-test gives ∣z∣=1.96. What is the p-value approximately?Two-sided p-value: both tails beyond |z|
Example 36
easy
At α=0.05, the decision when p=0.04 is:
Example 37
easy
A one-sided test gives upper-tail area 0.02 at the observed z. What is the one-sided p-value?One-sided (upper) p-value: upper-tail area is 0.02
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=0.04 and p=0.06, α=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=0.001 for a tiny effect (Δ=0.1 on a 100-point scale) with n=10000. Is the effect important?
Example 42
medium
A study reports p<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 as the per-test threshold. If overall α=0.05, what threshold should each test use?
Example 44
hard
An experiment yields p=0.049. A press release says 'the study proved the treatment works.' Why is this wrong?
Example 45
hard
Two independent studies report p=0.04 for the same hypothesis. Should we combine them by multiplying p-values?
Example 46
medium
True or false: a p-value above α means H0 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 after 1000 users. What does this tell us about whether B is better than A?
Example 49
hard
A researcher obtains p=0.03. Would the result be significant at α=0.05? At α=0.01?
Example 50
hard
A hypothesis test gives a p-value of 0.20. What decision would you make at α=0.10 and at α=0.05?