Hypothesis Tests

1. You are expected to know all of the intro stats-level material about hypothesis testing, including but not limited to:
   • setting up the null and alternative hypotheses for a claim
   • under a given pair of hypotheses, stating the meaning of a Type I or Type II Error
   • defining the Type I and Type II Error Rates of a test
   • defining the power of a test
   • performing a given hypothesis test using:
     1. a rejection region
     2. a confidence interval
     3. a \(P\)-value
2. Define the size of a hypothesis test.
3. Explain what it means for a hypothesis test to have (significance) level \(\alpha\).

Nonparametric Hypothesis Tests

1. Explain what it means for a distribution to be parametric.
2. Give examples of parametric distributions and state their parameters.
3. Explain the advantages of a nonparametric hypothesis test compared to a parametric hypothesis test.

The One-sample Sign Test

1. State the null and alternative hypotheses for the one-sample sign test.
2. State the test statistic for the one-sample sign test and its sampling distribution under the null hypothesis.
3. Give the rationale for the test statistic for the one-sample sign test.
4. Perform a one-sample sign test using SignTest() from the DescTools package.
5. Explain why there are a discrete set of \(P\)-values for the one-sample sign test.
6. Describe the general shape of the sampling distribution of \(P\)-values for a one-sample sign test when the null hypothesis is true.

Simulation

Statistical Characteristics of Hypothesis Tests

1. Given a population distribution and its mean, simulate a one-sample \(t\)-test from that population under the null or alternative hypothesis using t.test().
2. Given a population distribution and its median, simulate a one-sample sign test from that population under the null or alternative hypothesis using SignTest() from the DescTools package.
3. Use replicate() to simulate many \(P\)-values for a one-sample \(t\)-test or sign test under the null or alternative hypotheses.
4. Use ecdf() to approximate either the Type I Error Rate or Power of a hypothesis test from the simulated \(P\)-values.
5. Explain what the sampling distribution of \(P\)-values should look like under a point null hypothesis \(H_{0} : \theta = \theta_{0}\) when the null hypothesis is true.
6. Explain what the sampling distribution of \(P\)-values should look like under a point null hypothesis \(H_{0} : \theta = \theta_{0}\) when the null hypothesis is false.