A microbiologist is testing a water sample for E-coli....

A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, $H_{0}$, is: the sample does not contain $E$ coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is 0.012. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is 0.002. What is the power of this test?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to assess assumptions about a population by analyzing sample data. It involves setting up null and alternative hypotheses, determining error rates, and using statistical measures to make decisions about the presence or absence of an effect or characteristic.

Statistical Power

The power of a statistical test is the probability that the test correctly rejects a false null hypothesis, thereby detecting an effect or presence when it truly exists. It is calculated as one minus the probability of a Type II error, making it a key measure of a test's effectiveness.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the default state or claim that there is no effect or no difference, while the alternative hypothesis is the statement that contradicts the null, suggesting an effect or difference exists. This framework is used to decide whether the evidence supports a revision of the initial assumption.

Type I and Type II Errors

Errors in hypothesis testing fall into two types: Type I error (false positive), where the null hypothesis is incorrectly rejected when it is true, and Type II error (false negative), where the test fails to reject the null hypothesis when it is actually false. Understanding these errors is crucial for assessing the reliability of a test.

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