Practice Type I and Type II Errors in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Type I error (\alpha): rejecting H_0 when it is actually true (false positive). Type II error (\beta): failing to reject H_0 when it is actually false (false negative).
Think of a medical test. Type I error: the test says you have a disease when you don't (false alarm). Type II error: the test says you're healthy when you actually have the disease (missed detection). A smoke alarm that goes off when there's no fire is a Type I error; one that stays silent during a real fire is a Type II error. You can't eliminate both—reducing one tends to increase the other.
Example 1
mediumDefine Type I and Type II errors. A court uses 'innocent until proven guilty.' Identify which type of error corresponds to (a) convicting an innocent person, (b) acquitting a guilty person.
Example 2
hardA medical test has \alpha = 0.05 and \beta = 0.20 (Power = 0.80). If the true disease rate is 5% in the population: (a) in 100 truly diseased patients, how many will be missed? (b) In 1000 truly healthy patients, how many will get false positives?
Example 3
easyIf we set \alpha = 0.01 (stricter) instead of \alpha = 0.05, what happens to Type I error rate? What likely happens to Type II error rate?
Example 4
hardA factory quality test accepts shipments if sample defect rate is below 5%. H_0: defect rate ≤ 5% (accept). Type I: reject good shipment. Type II: accept bad shipment. Which error is more costly for the factory, and how should this affect choice of \alpha?