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Hypothesis Testing with One Sample

Introduction

LEARNING OBJECTIVES

Distinguish between Type I and Type II error and discuss the consequences of each.


KEY TAKEAWAYS

Key Points
  • A type I error occurs when the null hypothesis \left(\mathrm{H}_{0}\right) is true but is rejected.
  • The rate of the type I error is called the size of the test and denoted by the Greek letter \alpha (alpha).
  • A type II error occurs when the null hypothesis is false but erroneously fails to be rejected.
  • The rate of the type II error is denoted by the Greek letter \beta (beta) and related to the power of a test (which equals 1-\beta ).

Key Terms
  • type II error: Accepting the null hypothesis when the null hypothesis is false.
  • Type I error: Rejecting the null hypothesis when the null hypothesis is true.

The notion of statistical error is an integral part of hypothesis testing. The test requires an unambiguous statement of a null hypothesis, which usually corresponds to a default "state of nature" - for example "this person is healthy," "this accused is not guilty" or "this product is not broken. " An alternative hypothesis is the negation of null hypothesis (for example, "this person is not healthy," "this accused is guilty," or "this product is broken"). The result of the test may be negative, relative to null hypothesis (not healthy, guilty, broken) or positive (healthy, not guilty, not broken).

If the result of the test corresponds with reality, then a correct decision has been made. However, if the result of the test does not correspond with reality, then an error has occurred. Due to the statistical nature of a test, the result is never, except in very rare cases, free of error. The two types of error are distinguished as type I error and type II error. What we actually call type I or type II error depends directly on the null hypothesis, and negation of the null hypothesis causes type I and type II errors to switch roles.


Source: Boundless, https://courses.lumenlearning.com/boundless-statistics/chapter/hypothesis-testing-one-sample/
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