Understanding Type I and Type II Errors in Hypothesis Testing
School
University of Calgary**We aren't endorsed by this school
Course
STAT 423
Subject
Statistics
Date
Dec 10, 2024
Pages
2
Uploaded by BaronSquirrel4760
5. Types of Errors in Hypothesis TestingIn hypothesis testing, when researchers make decisions about the null hypothesis, there arealways risks of making incorrect conclusions. These risks are classified intotwo types oferrors:Type I errorandType II error. Understanding these errors is crucial for interpreting theresults of statistical tests and improving the reliability of conclusions drawn from data.There are two possible types of errors that can occur in hypothesis testing:1.Type I Error(False Positive):○Occurs when the null hypothesis is rejected when it is actually true.○Probability of Type I error is denoted by α\alphaα, the significance level.Example: Suppose a new drug is being tested for its effectiveness in reducing blood pressure.The null hypothesis (H₀) is that the drug has no effect on blood pressure. A Type I error wouldoccur if the test concludes that the drug does have an effect, when, in fact, it does not.2.Type II Error(False Negative):○Occurs when the null hypothesis is not rejected when the alternative hypothesisis actually true.○Probability of Type II error is denoted by β\betaβ.Example: Continuing with the drug example, the null hypothesis (H₀) is that the drug has noeffect. A Type II error would occur if the test concludes that the drug has no effect, when in factit does.Thepower of the testis the probability of correctly rejecting the null hypothesis when it is false.Power is related to the probability of a Type II error and is calculated as 1−β1 - \beta1−β.●Example: If a test has a power of 0.80, this means there is an 80% chance of correctlyrejecting the null hypothesis when it is false. This implies a 20% chance (β = 0.20) ofmaking a Type II error.Factors Influencing Power: The power of a test is influenced by several factors:1.Sample size (n): Larger sample sizes generally increase the power of a test becausethey reduce the variability of the sample estimates, making it easier to detect trueeffects.2.Effect size: The larger the effect or difference between groups, the higher the power.Small effects are harder to detect, so the power is lower for such tests.3.Significance level (α): A higher significance level (α) increases the power of a testbecause it makes it easier to reject the null hypothesis. However, this also increases thelikelihood of committing a Type I error, so there is a trade-off.4.Variability in the data: Lower variability (or standard deviation) in the data makes iteasier to detect true differences, increasing the power of the test.
Relationship Between Type I Error, Type II Error, and Power●Inverse Relationship: The probability of a Type I error (α) and the probability of a TypeII error (β) are inversely related. If you decrease the likelihood of a Type I error (e.g., bylowering the significance level α), the probability of making a Type II error (β) increases,and vice versa.●Trade-off: Researchers often face a trade-off between Type I and Type II errors. Forexample:○If you set a verystringent significance level(e.g., α = 0.01), you reduce thechances of making a Type I error, but you may increase the chance of making aType II error.○Conversely, if you set alarger significance level(e.g., α = 0.10), you increasethe chances of detecting an effect (i.e., increasing power) but also raise the riskof a Type I error.●Maximizing Power: Researchers aim to balance α and β in a way that optimizes thetest’s power. This involves considering theconsequencesof each type of error in thespecific context of the research. For example, in medical testing, aType I error(falsepositive) might lead to unnecessary treatment, while aType II error(false negative)might mean missing a potentially life-saving intervention.