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.
Showing a random 20 of 76 problems.
Example 1
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 2
medium
Statistical significance and practical significance can differ; a very small effect can still produce a small p-value with a ____ sample.
Example 3
hard
A study tests 40 outcomes at α=0.05 but all H0 are true. Expected number of 'significant' results is what?
Example 4
hard
Five tests at family-wise α=0.05 via Bonferroni: each test uses α= what?
Example 5
medium
Why does running many tests and reporting only the smallest p-value distort its meaning?
Example 6
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 7
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 8
easy
At α=0.05, a p-value of 0.20 leads to what decision?
Example 9
easy
True or false: smaller p-values mean larger effects.
Example 10
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 11
easy
At α=0.05, a p-value of 0.02 leads to what decision?
Example 12
easy
What threshold is most commonly used to call a p-value 'small'?
Example 13
medium
Why do statisticians prefer to report the exact p-value (e.g., 0.012) instead of just 'p < 0.05'?
Example 14
easy
Fill in: the p-value measures how ____ the data are, assuming the null is true.
Example 15
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 16
medium
Two studies report p =0.049 and p =0.051 at α=0.05. How different is the actual evidence?
Example 17
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 18
hard
Two independent studies report p=0.04 for the same hypothesis. Should we combine them by multiplying p-values?
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
medium
True or false: a p-value above α means H0 is true.