Chapter 20 Study Notes: Type I & II Errors and Statistical Power

The Nature of Decision Making in Statistics: The transcript notes that "nobody's perfect," meaning that even with an abundance of evidence, the wrong decision can still be made. Specifically, when performing a hypothesis test, mistakes can occur in two distinct ways.
Definitions of Error Categories: * Type I Error: Occurs when the null hypothesis ( $H_0$ ) is true, but we mistakenly reject it. * Type II Error: Occurs when the null hypothesis ( $H_0$ ) is false, but we fail to reject it.
Mnemonic for Classification: To keep the names straight, remember that statisticians start by assuming the null hypothesis is true; therefore, a Type I error is the "first" kind of error we could potentially make.
The Role of the P-value: Decisions are based on P-values, while the ultimate truth (noted as something "only God knows") determines whether those decisions are correct or erroneous.

Medical Disease Testing: * Hypotheses: The null hypothesis ( $H_0$ ) is usually the assumption that a person is healthy. The alternative hypothesis ( $H_A$ ) is that the individual has the disease being tested for. * Type I Error (False Positive): A healthy person is diagnosed with the disease. * Consequences: This may lead to an unnecessary chest X-ray. In cases like drug tests or diseases like AIDS, a false-positive result that is not kept confidential could have serious social or personal consequences. It may also mean the person must undergo further testing. * Type II Error (False Negative): An infected person is diagnosed as being disease-free. * Consequences: A sick patient goes untreated, which can be life-threatening depending on the severity of the illness.
Legal Context (Jury Trials): * Type I Error: The jury convicts a person who is actually innocent. * Type II Error: The jury fails to convict a person who is actually guilty.
Educational Context (Statistics Final Exam): * Hypotheses: Let $H_0$ be the assumption that the student has learned only $60\%$ of the material. * Type I Error: Passing a student who, in fact, learned less than $60\%$ of the material. * Type II Error: Failing a student who actually knew enough material to pass.
Evaluation of Seriousness: The determination of which error is "more serious" depends entirely on the specific situation, the associated costs, and the individual point of view.

Outcome Scenarios Table: * If the Truth is that $H_0$ is True: * Decision: Reject $H_0$ $\rightarrow$ Result: Type I Error. * Decision: Fail to reject $H_0$ $\rightarrow$ Result: OK (Correct decision). * If the Truth is that $H_0$ is False: * Decision: Reject $H_0$ $\rightarrow$ Result: OK (This is characterized as "Great" and "What we hope for"). * Decision: Fail to reject $H_0$ $\rightarrow$ Result: Type II Error (Characterized as "meh").
Defining Power: The Power of a test is defined as the ability to correctly detect a false $H_0$ .

Mechanism of Type I Errors: A Type I error happens when the null hypothesis is true, but the researcher has the "bad luck" to draw an unusual sample that leads to an unnecessary rejection.
The Alpha Level ( $\alpha$ ): The $\alpha$ -level is the probability of making a Type I error.
Comparing Alpha Thresholds: * Using $\alpha = 0.10$ : Researchers reject the null hypothesis more easily, but this leads to a higher chance of committing a Type I error. * Using $\alpha = 0.01$ : This creates a higher threshold for rejection, but results in a higher chance of a Type II error.
Reducing Error Rates: The transcript explicitly states that the "only way to reduce both [Type I and Type II errors] is to increase the sample size."

Activity Type I and Type II Errors: The material recommends viewing an animated exploration of these errors as a "good backup for the reading in this section."