Foundations — Unit Review
Unit 5 supplies the machinery underneath everything else: the rules of probability, the distributions that model random outcomes, and the power/sample-size ideas that decide whether a study can detect an effect at all. This review connects those pieces and drills two judgments — which distribution models this situation? and what makes a study powerful?
The big picture
- Probability is the language of chance. Probability rules (complements, unions, conditional probability, independence) let you reason about uncertain events, and random variables and expected value summarize a random process by its long-run average and spread.
- Distributions are the standard models. The binomial counts successes in a fixed number of independent yes/no trials; the normal models continuous, bell-shaped measurements — and, via the earlier mathematical-models idea, approximates many statistics in large samples.
- Power and design. Type II error, statistical power, and sample size ask the planning question: given the choices, how likely is the study to detect a real effect — and how many observations does that take?
Which distribution models this?
| Situation | Model | You compute |
|---|---|---|
| Count of successes in \(n\) independent yes/no trials, constant \(p\) | binomial | \(P(X = k)\), mean \(np\), SD \(\sqrt{np(1-p)}\) |
| A continuous, bell-shaped measurement | normal | areas/probabilities via z-scores |
| A count with large \(n\) (many successes & failures) | normal approximation to the binomial | area under a normal curve |
The binomial needs four things: a fixed number of trials \(n\), two outcomes per trial, independent trials, and a constant success probability \(p\). If any fails (e.g. sampling without replacement from a small population), the binomial model doesn’t apply.
What makes a study powerful?
Power = the probability of detecting a real effect = \(1 - \beta\) (where \(\beta\) is the Type II error rate). Power goes up when:
- the sample size \(n\) is larger,
- the true effect is larger,
- the discernibility level \(\alpha\) is larger (a more lenient bar), or
- the data are less variable.
Planning a study is choosing \(n\) to reach a target power (often 0.80) for an effect worth detecting — the job of the sample-size calculation.
Key ideas to carry forward
Complement / union / intersection · conditional probability · independence · random variable · expected value · binomial (\(n\), \(p\)) · normal model · z-score · normal approximation to the binomial · Type I error (\(\alpha\)) · Type II error (\(\beta\)) · power (\(1-\beta\)) · effect size · sample-size planning.
Common pitfalls
- Using the binomial when trials aren’t independent or \(p\) isn’t constant (e.g. drawing without replacement from a small group).
- Confusing \(P(A \mid B)\) with \(P(B \mid A)\) — conditional probabilities are directional.
- Treating “independent” and “mutually exclusive” as the same — they’re different ideas (mutually exclusive events with nonzero probability are not independent).
- Thinking a non-significant result proves no effect — it may just reflect low power (too small an \(n\)).
- Raising power by raising \(\alpha\) has a cost — it increases the Type I error rate.
Exercises
The Unit 5 skills are choosing the right distribution and reasoning about power. Answers are provided so you can check your work.
Binomial or normal? For each, name the model you’d use.
- The number of heads in 20 fair coin flips.
- The heights of adult women, modeled as bell-shaped with a known mean and SD.
- The number of defective items in a random sample of 50, each defective with probability 0.03.
- A standardized test score, used to find the percentile of a 1300.
Show answer
a. binomial (\(n=20\), \(p=0.5\)). b. normal. c. binomial (\(n=50\), \(p=0.03\)). d. normal (z-score → percentile).
Do the binomial conditions hold? You draw 5 cards from a standard deck without replacement and count the aces. Is the count binomial? Why or why not?
Show answer
No. Without replacement, the probability of an ace changes from draw to draw and the trials are not independent — two of the binomial requirements fail. (With replacement, it would be binomial.)
What raises power? A team’s study has only 60% power to detect the effect they care about. List three distinct changes that would raise the power, and one cost to watch for.
Show answer
Raise power by: (1) increasing the sample size, (2) reducing variability (better measurement/design), or (3) raising \(\alpha\) (a more lenient threshold). Watch the cost: raising \(\alpha\) increases the Type I error rate, so the usual lever is a larger \(n\). (Detecting a larger true effect also gives more power, but the effect size isn’t ours to choose.)
Type I vs. Type II. A drug trial sets \(\alpha = 0.05\). In plain terms, what is the Type I error here, what is the Type II error, and which one does \(\alpha\) control?
Show answer
Type I: concluding the drug works when it truly doesn’t (false positive) — its rate is controlled by \(\alpha\) (here 0.05). Type II: concluding the drug doesn’t work when it truly does (false negative) — its rate is \(\beta\), reduced mainly by a larger sample. Power is \(1-\beta\).
Conditional probability. 1% of a population has a disease. A test is positive in 90% of sick people and in 8% of healthy people. Is \(P(\text{disease} \mid \text{positive})\) large or small, and why does intuition often get this wrong?
Show answer
It’s small (roughly 10%): because the disease is rare, the many false positives from the 99% healthy population outnumber the true positives. Intuition conflates \(P(\text{positive} \mid \text{disease}) = 0.90\) with \(P(\text{disease} \mid \text{positive})\) — the reversed conditional — which is the base-rate fallacy.