Section 9.1: Tests of a Hypothesis Based on a Single Sample

1. Define the Type II Error Rate for a hypothesis test.
2. Recognize and use the convention of denoting the Type I Error rate by \(\alpha\) and the Type II Error Rate by \(\beta\).
3. For a given hypothesis test and a specified alternative value, compute the Type I and Type II Error Rates for the hypothesis test.
4. Define the power of a hypothesis test, and relate the power to the Type II Error Rate of the test.
5. Define the effect size \(\Delta\) for a hypothesis test for the population mean of a Gaussian population.
6. State how the power of a hypothesis changes with:
   • The sample size \(n\)
   • The effect size \(\Delta\)
   • The significance level \(\alpha\)

Section 9.2: Tests About a Population Mean

1. Use the pwr.norm.test function from the R package pwr to perform power calculations for hypothesis tests for the population mean of a Gaussian population.
   • Find \(n\) given \(\mu_{0}, \mu_{a}, \sigma, \alpha,\) and \(1 - \beta\).
   • Find \(1 - \beta\) given \(\mu_{0}, \mu_{a}, \sigma, \alpha,\) and \(n\).
2. Determine the distribution of the \(Z\)-statistic for a given hypothesis test for the population mean of a Gaussian population when \(\mu = \mu_{a} \neq \mu_{0}\).

Section 9.4: \(P\)-values

1. Define the \(P\)-value for an observed test statistic.
2. Compute the \(P\)-value for the observed test statistic in a hypothesis test for:
   • The population mean of a Gaussian population where \(\sigma\) is known.
   • The population mean of a Gaussian population where \(\sigma\) is unknown.
   • The success probability / population proportion for a binomial experiment.
3. Use a \(P\)-value to perform a hypothesis test at a significance level \(\alpha\).
4. (In the best of all possible worlds:) Avoid common mis-definitions and mis-interpretations of the \(P\)-value.