You flip a coin 100 times and get 60 heads. Your friend says the coin is biased.

• What's the null hypothesis?
• What would the null distribution look like? (Hint: what's the expected number of heads for a fair coin?)
• Is 60/100 surprising enough to reject fairness? What about 55/100? 90/100?
• How would your answer change if you only flipped 10 times and got 6 heads?

A pharmaceutical company tests a new drug on 500,000 patients. They find it reduces headache duration by an average of 0.3 minutes ($p < 0.0001$, $d = 0.01$).

• Is this result statistically significant? Is it meaningful?
• Would you take this drug? Would you prescribe it?
• What if the same drug reduced headache duration by 45 minutes in a study of 12 people ($p = 0.08$, $d = 1.2$)?
• Which study tells you more about whether the drug works?

A researcher wants to know if a mindfulness intervention reduces test anxiety.

Design A: She recruits 40 students. Half do mindfulness training; half don't. All take an anxiety questionnaire.

Design B: She recruits 20 students. They all take the anxiety questionnaire, then do 4 weeks of mindfulness training, then take the questionnaire again.

• Which design uses an independent-samples $t$-test? Which uses paired?
• What are the advantages and disadvantages of each design?
• Design B finds $p = 0.02$, but anxiety dropped by only 1 point on a 50-point scale. What do you conclude?

A sleep researcher assigns participants to sleep 4, 6, or 8 hours per night for a week, then measures reaction time. ANOVA yields $F(2, 57) = 12.4$, $p < 0.001$, $\eta^2 = 0.30$.

• What's the null hypothesis?
• We know the groups differ — but do we know if 4 hours is worse than 6? Or only that 4 is worse than 8?
• $\eta^2 = 0.30$ means sleep duration explains 30% of the variance in reaction time. Is that a lot? What explains the other 70%?
• Could you answer this question with three separate $t$-tests instead? Why is that problematic?

Consider these three findings:

1. Ice cream sales and drowning deaths are positively correlated ($r = 0.85$)
2. A student's shoe size correlates with reading ability ($r = 0.70$) in a sample of kids aged 5--15
3. Countries with more Nobel Prize winners per capita also consume more chocolate per capita ($r = 0.79$)

For each:

• Does the correlation reflect a causal relationship?
• What's the lurking variable?
• How would you design a study to test whether the relationship is causal?

Suppose a regression predicting exam scores from sleep and caffeine intake gives:

• Sleep: $\beta = 3.98$ (each extra hour of sleep → ~4 more exam points)
• Caffeine: $\beta = 0.01$ (essentially zero)

But the correlation between caffeine and exam scores is $r = -0.30$.

• Why does caffeine "disappear" in the regression but show up in the correlation?
• What does this tell you about the relationship between caffeine, sleep, and exam scores?
• When would you use correlation vs. regression to answer a research question?

A university surveys 200 students about their preferred study location (library, dorm, café) and their major (STEM, humanities, social science). They find $\chi^2(4) = 15.3$, $p = 0.004$, $V = 0.20$.

• What's the null hypothesis?
• What would the expected frequencies look like if major and location were truly independent?
• The result is significant — but can we say "STEM students prefer the library"? What additional analysis would we need?
• Could you use a $t$-test here instead? Why or why not?

A college claims that 85% of its graduates find employment within 6 months. A skeptical journalist surveys 80 recent graduates and finds that 60 (75%) are employed.

• What's the null hypothesis?
• Is 75% vs. 85% a meaningful difference, or could it be sampling variability?
• The binomial test gives $p = 0.028$. What do you conclude?
• Now imagine a different college claims 50% employment and you observe 44/80 (55%). The binomial test gives $p = 0.43$. Same question: what do you conclude?
• How does sample size affect your ability to detect a difference from the claimed rate?

For each scenario, identify the appropriate test and state the null hypothesis:

1. You want to know if students who sit in the front row score higher than those in the back
2. You're testing whether coffee preference (espresso, drip, cold brew) is related to personality type (introvert vs. extrovert)
3. A teacher measures test scores before and after switching to a new textbook, for the same 30 students
4. You want to predict final exam score from GPA, attendance rate, and hours of sleep
5. In a sample of 50 coin flips by a magician, 34 land on heads