List The Four Possible Results Of Decision Combinations ✓ Solved

List the four possible results of the combinations of decisions

List the four possible results of the combinations of decisions and true states of nature for a test of hypothesis.

Paper For Above Instructions

In statistical hypothesis testing, the four possible outcomes of combining decisions (rejecting or not rejecting the null hypothesis) with true states of nature (true null hypothesis or true alternative hypothesis) are fundamental to understanding the effectiveness and reliability of hypothesis testing. These outcomes are as follows:

1. True Positive (Correct Decision)

The first outcome is a true positive, which occurs when the null hypothesis (H0) is false, and the decision is made to reject it. In this scenario, the test correctly identifies the existence of a significant effect or relationship. For example, if a new drug is intended to lower blood pressure, and the statistical analysis shows a significant reduction when it actually exists, this is a true positive. The probability of making a true positive decision is represented by power (1 - β), where β denotes the probability of making a Type II error.

2. True Negative (Correct Decision)

The second outcome is a true negative, which occurs when the null hypothesis is true, and the decision is made not to reject it. In this case, the test correctly identifies the absence of a significant effect or difference. This reflects a scenario where the hypothesis holds true—such as a new educational program that does not yield improved student performance when statistically tested. The probability of achieving a true negative is denoted by (1 - α), where α signifies the probability of committing a Type I error.

3. False Positive (Type I Error)

4. False Negative (Type II Error)

The fourth outcome is referred to as a false negative or Type II error, which occurs when the null hypothesis is false, but the decision is made not to reject it. This outcome reflects an inability of the test to detect an effect or a difference that truly exists. For instance, if a new treatment for a disease is less effective than a standard treatment, yet the statistical analysis fails to reject the null hypothesis, this results in a false negative. The probability of committing a Type II error is represented by β.

Summary

In summary, the four possible outcomes of hypothesis testing can lead to accurate or inaccurate conclusions about the population from which a sample is drawn. True positives and true negatives reflect correct decisions based on the data, while false positives and false negatives represent inaccuracies. Understanding these four outcomes is crucial as they underscore the importance of selecting appropriate significance levels, adequately powering a study, and interpreting the implications of statistical testing in research and practical applications.

Conclusion

In real-world applications, these four possible outcomes are highly significant as they influence decisions in fields ranging from medicine to education, and even business analytics. Researchers and practitioners must be aware of these concepts to minimize errors and improve the reliability of their conclusions, thus facilitating better decision-making.

References

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