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.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

challenge
Explain why, holding the sample size fixed, you cannot simultaneously reduce both α\alpha and β\beta.

Example 3

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 4

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 5

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 6

easy
A drug actually works but the test fails to reject the 'no effect' null. Which error?

Example 7

easy
The probability of a Type I error is denoted by which Greek letter?

Example 8

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?

Example 9

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 10

easy
A medical test says a healthy person has a disease. Which error is this?

Example 11

easy
A Type I error is sometimes called a ____ positive.

Example 12

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

Example 13

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 14

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 15

medium
A juror convicts an innocent defendant. With H0:H_0: 'innocent', which error type is this?

Example 16

easy
Statistical power is defined as which quantity?

Example 17

easy
True or false: α+β=1\alpha + \beta = 1 in general.

Example 18

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 19

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

Example 20

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