Type I and Type II Errors Examples: 47 Problems with Answers
Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Type I and Type II Errors.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Math.
Concept Recap
Type I error (α): rejecting H0 when it is actually true (false positive). Type II error (β): failing to reject H0 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.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A Type I error rejects a true null (false positive); a Type II error fails to reject a false null (false negative).
Common stuck point:The procedure for type i and type ii errors is the easy part; the trap is swapping Type I and Type II. Asking "Am I classifying a wrong decision by comparing what the test concluded against what is actually true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I classifying a wrong decision by comparing what the test concluded against what is actually true?
Worked Examples
Example 1
medium
Define 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.
Answer
(a) Convicting innocent = Type I error. (b) Acquitting guilty = Type II error.
First step
1
Type I error (false positive, α): reject H0 when H0 is true; probability = α
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
A medical test has α=0.05 and β=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
medium
A test has α=0.05 and power 0.80 for a specific alternative. (a) Find β. (b) Among 100 truly different samples, roughly how many will the test miss?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
If we set α=0.01 (stricter) instead of α=0.05, what happens to Type I error rate? What likely happens to Type II error rate?
Example 2
hard
A factory quality test accepts shipments if sample defect rate is below 5%. H0: 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 α?
Example 3
easy
Rejecting H0 when H0 is actually true is which type of error?
Example 4
easy
Failing to reject H0 when H0 is actually false is which type of error?
Example 5
easy
A medical test says a healthy person has a disease. Which error is this?
Example 6
easy
A medical test says a sick person is healthy. Which error is this?
Example 7
easy
The probability of a Type I error is denoted by which Greek letter?
Example 8
easy
The probability of a Type II error is denoted by which Greek letter?
Example 9
easy
If α=0.05, what is the probability of a Type I error when H0 is true?
Example 10
easy
Statistical power is defined as which quantity?
Example 11
medium
A spam filter blocks a legitimate email (treating 'not spam' as H0). Which error, and what is the trade-off if you make the filter more aggressive?
Example 12
medium
A factory tests H0: 'batch is good' vs Ha: 'batch is defective'. Shipping a defective batch (failing to reject when false) is which error?
Example 13
medium
Decreasing α from 0.05 to 0.01 (same n) has what effect on the Type II error rate β?
Example 14
medium
A test has β=0.20. What is its power?
Example 15
medium
Which single change increases power WITHOUT increasing the Type I error rate?
Example 16
medium
A juror convicts an innocent defendant. With H0: 'innocent', which error type is this?
Example 17
medium
Does α=0.05 mean 'there is a 5% chance my conclusion is wrong'? Explain the correct meaning.
Example 18
challenge
Explain why, holding the sample size fixed, you cannot simultaneously reduce both α and β.
Example 19
challenge
A screening test for a rare disease (prevalence 1%) has α=0.05 and power 0.90. Among 10000 people, roughly how many false positives occur?
Example 20
challenge
Two tests are proposed. Test A: α=0.10, power 0.95. Test B: α=0.01, power 0.70. For detecting a deadly disease where missing a case is far worse than a false alarm, which test is preferable and why?
Example 21
medium
A fire alarm sounds when there is no fire. Treating 'no fire' as H0, which error is this?
Example 22
medium
A test has α=0.05 and β=0.10. Find the power and the probability of a false positive when H0 is true.
Example 23
easy
If β=0.25, what is the power of the test?
Example 24
easy
If α=0.10, what is the long-run rate of Type I errors when H0 is true?
Example 25
easy
A drug actually works but the test fails to reject the 'no effect' null. Which error?
Example 26
medium
Suppose 1000 people with no disease are screened by a test with α=0.05. About how many false positives appear?
Example 27
medium
If a researcher increases α from 0.05 to 0.10 (sample size and effect size fixed), what happens to power?
Example 28
medium
True or false: increasing the effect size (difference between truth and H0 value) increases power.
Example 29
medium
True or false: increasing the population standard deviation σ (everything else fixed) increases power.
Example 30
medium
A judge convicts an innocent defendant (H0: innocent). Identify the error type and the conceptually 'safer' adjustment.
Example 31
medium
H0: a fire detector says 'no fire.' The detector misses an actual fire. Which error?
Example 32
medium
Among 200 healthy people screened with α=0.02, how many false positives are expected?
Example 33
medium
A test has α=0.05 and power 0.70. Compute β and the chance of a correct decision when H0 is true.
Example 34
medium
A test rejects H0. Which error is even possible at this point: Type I, Type II, or neither?
Example 35
medium
A test fails to reject H0. Which error is possible at this point?
Example 36
hard
True or false: increasing the sample size n can decrease BOTH α (in practice, by allowing a stricter threshold at the same power) and β.
Example 37
hard
Disease prevalence is 2%. A test has α=0.05 and power 0.80. Out of 10000 screened, roughly how many TRUE positives are expected?
Example 38
hard
Using the previous setup (prevalence 2%, α=0.05, 10000 screened), how many false positives are expected?
Example 39
hard
Using the previous setup (prevalence 2%, α=0.05, power 0.80, 10000 screened), compute the positive predictive value P(disease∣test positive).
Example 40
hard
Holding α fixed, which of these does NOT change power: (a) sample size n, (b) significance level α, (c) effect size, (d) population σ?
Example 41
hard
Test A: α=0.05, power 0.50. Test B: α=0.05, power 0.90. Which test is preferable for detecting a real effect, and why?
Example 42
hard
True or false: a test with α=0.05 that fails to reject H0 proves the null hypothesis is true.
Example 43
challenge
Multiple testing: 20 independent tests are run at α=0.05, each on a TRUE null. What is the probability of at least one false positive?
Example 44
challenge
A quality test has α=0.05. The factory wants to control the family-wise Type I rate over 10 independent tests at 0.05. Using the Bonferroni correction, what individual α should each test use?