Hypothesis Testing with One Sample

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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A type I error occurs when the null hypothesis $\left(\mathrm{H}_{0}\right)$ is true but is rejected. It is asserting something that is absent, a false hit. A type I error may be compared with a so-called false positive (a result that indicates that a given condition is present when it actually is not present) in tests where a single condition is tested for. A type I error can also be said to occur when we believe a falsehood. In terms of folk tales, an investigator may be "crying wolf" without a wolf in sight (raising a false alarm). $\mathrm{H}_{0}$: no wolf.

The rate of the type I error is called the size of the test and denoted by the Greek letter $\alpha$ (alpha). It usually equals the significance level of a test. In the case of a simple null hypothesis, $\alpha$ is the probability of a type I error. If the null hypothesis is composite, $\alpha$ is the maximum of the possible probabilities of a type I error.

A type II error occurs when the null hypothesis is false but erroneously fails to be rejected. It is failing to assert what is present, a miss. A type II error may be compared with a so-called false negative (where an actual "hit" was disregarded by the test and seen as a "miss") in a test checking for a single condition with a definitive result of true or false. A type II error is committed when we fail to believe a truth. In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"). Again, $\mathrm{H}_{0}:$ no wolf.

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 $1-\beta$).