Section 12.6 Summary
Subsection 12.6.1 Glossary
- Hypothesis Testing
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A method for determining between two hypothesis, with one of the two being the currently accepted hypothesis. A determination is based on the value of the test statistic. The probability of falsely rejecting the currently accepted hypothesis is the significance level of the test.
- Null Hypothesis (\(H_0\))
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A sub-collection that emerges in response to the situation when the phenomena is absent. The established scientific theory that is being challenged. The hypothesis which is worse to erroneously reject.
- Alternative Hypothesis (\(H_1\))
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A sub-collection that emerges in response to the presence of the investigated phenomena. The new scientific theory that challenges the currently established theory.
- Test Statistic
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A statistic that summarizes the data in the sample in order to decide between the two alternative.
- Rejection Region
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A set of values that the test statistic may obtain. If the observed value of the test statistic belongs to the rejection region then the null hypothesis is rejected. Otherwise, the null hypothesis is not rejected.
- Type I Error
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The null hypothesis is correct but it is rejected by the test.
- Type II Error
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The alternative hypothesis holds but the null hypothesis is not rejected by the test.
- Significance Level
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The probability of a Type I error. The probability, computed under the null hypothesis, of rejecting the null hypothesis. The test is constructed to have a given significance level. A commonly used significance level is 5%.
- Statistical Power
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The probability, computed under the alternative hypothesis, of rejecting the null hypothesis. The statistical power is equal to 1 minus the probability of a Type II error.
- \(p\)-value
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A form of a test statistic. It is associated with a specific test statistic and a structure of the rejection region. The \(p\)-value is equal to the significance level of the test in which the observed value of the statistic serves as the threshold.
Subsection 12.6.2 Discuss in the forum
In statistical thinking there is a tenancy towards conservatism. The investigators, enthusiastic to obtain positive results, may prefer favorable conclusions and may tend over-interpret the data. It is the statistician’s role to add to the objectivity in the interpretation of the data and to advocate caution.
On the other hand, the investigators may say that conservatism and science are incompatible. If one is too cautious, if one is always protecting oneself against the worst-case scenario, then one will not be able to make bold new discoveries.
Which of the two approach do you prefer?
When you formulate your answer to this question it may be useful to recall cases in your past in which you where required to analyze data or you were exposed to other people’s analysis. Could the analysis benefit or be harmed by either of the approaches?
For example, many scientific journal will tend to reject a research paper unless the main discoveries are statistically significant (\(p\)-value \(<\) 5%). Should one not publish also results that show a significance level of 10%?
Subsection 12.6.3 Formulas
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Test Statistic for Expectation: \(t = (\bar x - \mu_0)/ (s/\sqrt{n})\text{.}\)
