SOLID STATE PRESS
← Back to catalog
P-Values Explained cover
Buy on Amazon
US list price $2.99
Mathematics

P-Values Explained

Null Hypotheses, Significance Thresholds, and What 0.05 Actually Means — A TLDR Primer

Statistics class throws the term "p-value" at you, your textbook buries the explanation under pages of theory, and you still walk out of the chapter unsure what 0.05 actually means. This guide fixes that.

**P-Values Explained** is a concise, no-filler primer built for high school and early college students who need to understand hypothesis testing without getting lost in dense notation. It covers everything from the ground up: what a p-value really measures (hint: it is not the probability your hypothesis is true), how to set up a proper hypothesis test with H₀ and H₁, how to compute p-values using z-tests and t-tests, and — critically — how to read results without falling into the misinterpretations that trip up even working scientists.

The guide also covers Type I and Type II errors, statistical power, and why sample size matters more than most students realize. A final section tackles p-hacking, the replication crisis, and why the 0.05 threshold is under serious scrutiny in modern research — context that turns a textbook formula into something you can actually think with.

Short by design, tight on examples, and honest about where the common mistakes live. If you are prepping for an AP Statistics exam, a college intro stats course, or just trying to make sense of a confusing concept your instructor glossed over, this is the place to start.

Scroll up and grab your copy.

What you'll learn
  • Define null and alternative hypotheses and state them for a real problem
  • Compute a p-value from a z-test or t-test by hand and with a table
  • Interpret a p-value correctly and identify common misinterpretations
  • Explain the role of significance levels, Type I and Type II errors, and statistical power
  • Recognize the limits of p-values, including p-hacking and the replication crisis
What's inside
  1. 1. What a P-Value Actually Is
    Introduce the p-value as a probability about data assuming the null hypothesis is true, using a coin-flip example.
  2. 2. Setting Up a Hypothesis Test
    Walk through the structure of a hypothesis test: stating H0 and H1, choosing a significance level, and picking one- vs two-tailed.
  3. 3. Computing P-Values: Z-Tests and T-Tests
    Show how to compute p-values from z-scores and t-scores with worked examples using tables and calculator output.
  4. 4. How to Read a P-Value Without Being Wrong
    Address the most common misinterpretations of p-values and state precisely what a p-value does and does not say.
  5. 5. Errors, Power, and the Cost of Thresholds
    Explain Type I and Type II errors, the meaning of statistical power, and how sample size changes what counts as significant.
  6. 6. P-Hacking, Replication, and Why 0.05 Is Under Fire
    Cover the real-world abuse of p-values, the replication crisis, and modern alternatives like confidence intervals and Bayesian reasoning.
Published by Solid State Press · June 2026
P-Values Explained cover
TLDR STUDY GUIDES

P-Values Explained

Null Hypotheses, Significance Thresholds, and What 0.05 Actually Means — A TLDR Primer
Solid State Press

P-Values Explained

Null Hypotheses, Significance Thresholds, and What 0.05 Actually Means — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What a P-Value Actually Is
  2. 2 Setting Up a Hypothesis Test
  3. 3 Computing P-Values: Z-Tests and T-Tests
  4. 4 How to Read a P-Value Without Being Wrong
  5. 5 Errors, Power, and the Cost of Thresholds
  6. 6 P-Hacking, Replication, and Why 0.05 Is Under Fire
Chapter 1

What a P-Value Actually Is

Flip a coin ten times and get eight heads. Did you just catch a cheating coin, or did ordinary luck produce that result? The answer you want is not "what are the odds that coin is fair?" — it is something more precise: if the coin were fair, how likely would it be to produce a result this extreme? That probability is the p-value.

A p-value is a number between 0 and 1. It measures how surprising your data would be in a world where nothing unusual is going on — specifically, in a world where the null hypothesis is true. The null hypothesis ($H_0$) is the default claim, the assumption that there is no real effect, no difference, no bias. In the coin example, $H_0$ says the coin is fair: each flip has a 50% chance of landing heads. The p-value then asks: given that $H_0$ is true, how probable is it to observe data at least as extreme as what I actually got?

The phrase "at least as extreme" is doing real work there. The p-value is not the probability of getting exactly eight heads. It is the probability of getting eight heads, or nine heads, or ten heads — results that are equally or more inconsistent with a fair coin. This tail-end probability is what makes a p-value meaningful as a measure of surprise.

The Null Hypothesis and Its Counterpart

Every hypothesis test needs a second claim alongside $H_0$. The alternative hypothesis ($H_1$, sometimes written $H_a$) is what you suspect might actually be true. It is the claim you will accept if the data make the null look implausible. In the coin example, $H_1$ might be: the coin is biased toward heads (probability of heads $> 0.5$). You will set up $H_1$ carefully in Section 2 — for now, just notice that the p-value is calculated under the assumption that $H_0$ is true, not $H_1$.

A common mistake is to think the p-value measures how likely your hypothesis is. It does not. It measures how likely your data would be if a specific hypothesis ($H_0$) were true. That distinction is subtle but critical, and Section 4 will unpack it at length.

From Data to a Test Statistic

About This Book

If you are staring down a statistics hypothesis study guide for AP Statistics or a college intro stats course and the phrase "null hypothesis significance testing explained" still makes your eyes glaze over, this book is for you. It works equally well for a high school student needing p-value interpretation help before an exam and for a college freshman whose professor just assumed everyone retained it from high school.

This primer walks through what a p-value actually measures, how to set up a null hypothesis, when to use a z-test versus a t-test, and how to avoid the misreadings that trip up even working scientists. It covers type 1 and type 2 error, statistical power, and why the 0.05 threshold is increasingly contested. Think of it as a p-value explained for beginners — but without the condescension. Short by design, no filler.

Read straight through once to build the framework, then work every example as you go. The problem set at the end is your checkpoint for understanding statistical significance before the real test.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Continue reading on Amazon