\(X\) observed data. \(\phi_\alpha\) a family of tests functions, one for each level \(\alpha\).

We define the \(p\)-value \[ p = \min_\alpha \left[\phi_\alpha(X) > 0\right] \]

For tests that compare a statistic to a known distribution (Z-scores, T-scores, \(\chi^2\)-tests, …) the \(p\)-value can be calculated directly from the distribution function. Let \(z\) be the Z-score, distributed as \(\mathcal N(0,1)\). \[ p = \mathbb P(Z > z) = 1-\mathbb P(Z < z) = 1-\text{CDF}_{\mathcal N(0,1)}(z) \]

Very common is to require a particular threshold for statistical signficance: getting a small enough \(p\)-value is the difference between

• publishing or not
• getting a PhD or not
• getting hired or not
• getting a research grant or not
• getting tenure or not
• getting promotion or not

Each group will get an article.

1. Read the article carefully on your own.
2. Discuss the article within the group.
3. Describe your article to the rest of the group.

• https://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing
• https://www.americanscientist.org/article/the-statistical-crisis-in-science
• http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124