Segment 12. P-Value Tests
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Problems
To Calculate
1. What is the critical region for a 5% two-sided test if, under the null hypothesis, the test statistic is distributed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Student}(0,\sigma,4)} ? That is, what values of the test statistic disprove the null hypothesis with p < 0.05? (OK to use Python, MATLAB, or Mathematica.)
2. For an exponentially distributed test statistic with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} (under the null hypothesis), when is the the null hypothesis disproved with p < 0.01 for a one-sided test? for a two-sided test?
To Think About
1. P-value tests require an initial choice of a test statistic. What goes wrong if you choose a poor test statistic? What would make it poor?
2. If the null hypothesis is that a coin is fair, and you record the results of N flips, what is a good test statistic? Are there any other possible test statistics?
3. Why is it so hard for a Bayesian to do something as simple as, given some data, disproving a null hypothesis? Can't she just compute a Bayes odds ratio, P(null hypothesis is true)/P(null hypothesis is false) and derive a probability that the null hypothesis is true?
