Type I and Type II Errors Examples in Math

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\alpha): rejecting H0H_0 when it is actually true (false positive). Type II error (β\beta): failing to reject H0H_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.

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How to Use These Examples

  • 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\alpha): reject H0H_0 when H0H_0 is true; probability = α\alpha

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Example 2

hard
A medical test has α=0.05\alpha = 0.05 and β=0.20\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

medium
A test has α=0.05\alpha = 0.05 and power 0.800.80 for a specific alternative. (a) Find β\beta. (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\alpha = 0.01 (stricter) instead of α=0.05\alpha = 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%. H0H_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 α\alpha?

Example 3

easy
Rejecting H0H_0 when H0H_0 is actually true is which type of error?

Example 4

easy
Failing to reject H0H_0 when H0H_0 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\alpha = 0.05, what is the probability of a Type I error when H0H_0 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 H0H_0). Which error, and what is the trade-off if you make the filter more aggressive?

Example 12

medium
A factory tests H0:H_0: 'batch is good' vs Ha:H_a: 'batch is defective'. Shipping a defective batch (failing to reject when false) is which error?

Example 13

medium
Decreasing α\alpha from 0.050.05 to 0.010.01 (same nn) has what effect on the Type II error rate β\beta?

Example 14

medium
A test has β=0.20\beta = 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:H_0: 'innocent', which error type is this?

Example 17

medium
Does α=0.05\alpha = 0.05 mean 'there is a 5%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\alpha and β\beta.

Example 19

challenge
A screening test for a rare disease (prevalence 1%1\%) has α=0.05\alpha = 0.05 and power 0.900.90. Among 1000010000 people, roughly how many false positives occur?

Example 20

challenge
Two tests are proposed. Test A: α=0.10\alpha = 0.10, power 0.950.95. Test B: α=0.01\alpha = 0.01, power 0.700.70. 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 H0H_0, which error is this?

Example 22

medium
A test has α=0.05\alpha = 0.05 and β=0.10\beta = 0.10. Find the power and the probability of a false positive when H0H_0 is true.

Example 23

easy
If β=0.25\beta = 0.25, what is the power of the test?

Example 24

easy
If α=0.10\alpha = 0.10, what is the long-run rate of Type I errors when H0H_0 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\alpha = 0.05. About how many false positives appear?

Example 27

medium
If a researcher increases α\alpha from 0.050.05 to 0.100.10 (sample size and effect size fixed), what happens to power?

Example 28

medium
True or false: increasing the effect size (difference between truth and H0H_0 value) increases power.

Example 29

medium
True or false: increasing the population standard deviation σ\sigma (everything else fixed) increases power.

Example 30

medium
A judge convicts an innocent defendant (H0H_0: innocent). Identify the error type and the conceptually 'safer' adjustment.

Example 31

medium
H0H_0: 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\alpha = 0.02, how many false positives are expected?

Example 33

medium
A test has α=0.05\alpha = 0.05 and power 0.700.70. Compute β\beta and the chance of a correct decision when H0H_0 is true.

Example 34

medium
A test rejects H0H_0. Which error is even possible at this point: Type I, Type II, or neither?

Example 35

medium
A test fails to reject H0H_0. Which error is possible at this point?

Example 36

hard
True or false: increasing the sample size nn can decrease BOTH α\alpha (in practice, by allowing a stricter threshold at the same power) and β\beta.

Example 37

hard
Disease prevalence is 2%2\%. A test has α=0.05\alpha = 0.05 and power 0.800.80. Out of 10000 screened, roughly how many TRUE positives are expected?

Example 38

hard
Using the previous setup (prevalence 2%2\%, α=0.05\alpha = 0.05, 10000 screened), how many false positives are expected?

Example 39

hard
Using the previous setup (prevalence 2%2\%, α=0.05\alpha = 0.05, power 0.800.80, 10000 screened), compute the positive predictive value P(disease∣test positive)P(\text{disease} \mid \text{test positive}).

Example 40

hard
Holding α\alpha fixed, which of these does NOT change power: (a) sample size nn, (b) significance level α\alpha, (c) effect size, (d) population σ\sigma?

Example 41

hard
Test A: α=0.05\alpha = 0.05, power 0.500.50. Test B: α=0.05\alpha = 0.05, power 0.900.90. Which test is preferable for detecting a real effect, and why?

Example 42

hard
True or false: a test with α=0.05\alpha = 0.05 that fails to reject H0H_0 proves the null hypothesis is true.

Example 43

challenge
Multiple testing: 20 independent tests are run at α=0.05\alpha = 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\alpha = 0.05. The factory wants to control the family-wise Type I rate over 10 independent tests at 0.050.05. Using the Bonferroni correction, what individual α\alpha should each test use?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

hypothesis testingp value