📈 Intro to Statistics · Lesson 7 of 7
Hypothesis tests and p-values
Null and alternative, the p-value, significance and errors.
The logic
A hypothesis test is like a trial. The null hypothesis H₀ says “nothing is going on” (no effect, no difference) — innocent until proven guilty. The alternative Hₐ is what you suspect. We ask: if H₀ were true, how surprising would our data be?
The p-value
The p-value is the probability, assuming H₀ is true, of getting results at least as extreme as what we observed. A small p-value (data would be surprising under H₀) is evidence against H₀.
Example
A coin lands heads 62 times in 100 flips. H₀: the coin is fair (p = 0.5). Hₐ: it isn’t. If the coin were fair, getting 62 or more heads — or 38 or fewer — has a probability of about 0.02. That’s the p-value. Since 0.02 < 0.05, we reject H₀: there is convincing evidence the coin is not fair.
Decisions and errors
| Significance level α | Usually 0.05 — reject H₀ if p < α |
|---|---|
| Type I error | Rejecting a true H₀ (a false alarm) — probability α |
| Type II error | Failing to reject a false H₀ (a missed effect) |
| “Fail to reject” | Not the same as proving H₀ true |
Statistical vs. practical significance
With a huge sample, a tiny, unimportant difference can be “statistically significant”. Always report the size of the effect (and a confidence interval), not just p < 0.05.
✏️ Practice: your turn
- Write H₀ and Hₐ for: “Does a new study app raise average quiz scores?”
- Explain a p-value of 0.30 in one sentence
- Describe a Type I and a Type II error for a medical test
- Find a study in the news and check whether it reports effect size
✏️ Scientific calculator with a statistics tab (mean, median, standard deviation)
🧠 Quick quiz
Answer all 4 questions correctly to mark this lesson as done.
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All lessons in Intro to Statistics
- 🎲 1. Data, variables and samplingCategorical vs. quantitative, populations, samples and bias
- 🎯 2. Mean, median and modeMeasures of center, and which one to use when data are skewed
- ↔️ 3. Spread: range, IQR and standard deviationHow spread out the data are, and how to read a box plot
- 🔔 4. The normal curve and z-scoresThe 68–95–99.7 rule and comparing values on different scales
- 📉 5. Correlation and regressionScatterplots, r, the line of best fit — and causation
- 🎯 6. Confidence intervalsEstimate ± margin of error, and what “95% confident” means
- ⚗️ 7. Hypothesis tests and p-valuesNull and alternative, the p-value, significance and errors