Practice P-Value in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The probability of observing a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis H0H_0 is true.

The p-value answers: 'If nothing special is going on (H0H_0 is true), how surprising is my data?' A tiny p-value means the data would be very rare under H0H_0, so maybe H0H_0 is wrong. Think of it like this: you flip a coin 100 times and get 92 heads. If the coin is fair, the chance of that happening is astronomically small (tiny p-value)—so you'd conclude the coin is probably not fair.

Showing a random 20 of 50 problems.

Example 1

challenge
Show why, under H0H_0, P(p-value≤α)=αP(\text{p-value} \le \alpha) = \alpha, and connect this to the Type I error rate.

Example 2

easy
Define in words what a p-value measures.

Example 3

medium
Fill in: if the p-value is below α\alpha, the result is called statistically ____.

Example 4

easy
A two-sided test has one-tail area 0.0150.015. What is the p-value?

Example 5

medium
A p-value of exactly α\alpha leads to what conclusion under the convention p≤αp \le \alpha?

Example 6

easy
True or false: the p-value is the probability that H0H_0 is true.

Example 7

medium
A two-sided test produces z=−1.96z = -1.96 with P(Z<−1.96)=0.025P(Z < -1.96) = 0.025. Find the p-value.

Example 8

medium
A test gives p=0.30p = 0.30. A student writes 'there is a 30%30\% chance H0H_0 is true.' Correct the interpretation.

Example 9

easy
True or false: a p-value of 0.00010.0001 proves the alternative hypothesis is true.

Example 10

medium
A study reports p=0.045p = 0.045. A reader claims 'there is a 4.5%4.5\% chance the treatment has no effect.' Why is this wrong?

Example 11

challenge
A test statistic has p-value 0.040.04. The researcher then collects more data hoping to 'confirm' significance, stopping when p<0.05p < 0.05 again. Why does this invalidate the p-value?

Example 12

easy
Fill in the blank: A small p-value is evidence ____ the null hypothesis.

Example 13

challenge
Prove: if the test statistic TT has a continuous distribution under H0H_0, then the p-value P=P(T≥Tobs∣H0)P = P(T \ge T_{\text{obs}} \mid H_0) is uniformly distributed on (0,1)(0,1) when H0H_0 holds.

Example 14

easy
A right-tailed test produces z=0.5z = 0.5. Use P(Z>0.5)=0.3085P(Z > 0.5) = 0.3085 to find the p-value and decide at α=0.05\alpha = 0.05.

Example 15

hard
A t-test with df=9df = 9 produces t=2.262t = 2.262, the critical value for α=0.05\alpha = 0.05 two-sided. Find the p-value.

Example 16

easy
A one-tailed test has z=1.7z=1.7. Find the p-value and determine if we reject H0H_0 at α=0.05\alpha=0.05.

Example 17

hard
A study reports 'no significant difference, p=0.32p = 0.32.' Can we conclude there is no difference between groups?

Example 18

hard
Correct the following misconceptions about p-values: (a) 'p=0.03 means there's a 3% chance H₀ is true.' (b) 'p=0.03 means the effect is large.'

Example 19

challenge
A journal publishes only p<0.05p < 0.05 results. Explain why published effect sizes tend to be overstated (the 'winner's curse').

Example 20

medium
A hypothesis test produces z=2.3z=2.3 for a two-tailed test. Calculate the p-value and interpret it at both α=0.05\alpha=0.05 and α=0.01\alpha=0.01.
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