Inferential Statistics

1. State the three main tasks of inferential statistics.
2. State the three procedures used to perform the three main tasks of inferential statistics.
3. Explain why the outcomes of inferential procedures are themselves random.
4. Explain what it means for an inferential procedure to have desirable frequency properties.

Simulation

Big Picture

1. Explain the analogy between \(X_{1}, X_{2}, \ldots, X_{n} \stackrel{\text{iid}}{\sim} D\) (in math) and Xs <- rD(n) (in R).

Sampling Distributions via Simulation

1. Given a population distribution and a sample statistic that can be calculated using R, construct a single realization of the sample statistic in R.
2. Use replicate() to repeatedly generate a random sample and calculate an associated sample statistic.
3. Compare the empirical sampling distribution from simulation to a known sampling distribution from theory.

Fisher’s Confidence Interval for \(\rho\)

1. State Fisher’s confidence interval for the population correlation \(\rho\) of a bivariate Gaussian based on his \(Z\)-transformation of the sample correlation \(R\).
2. Compute the hyperbolic functions \(\sinh x\), \(\cosh x\), and \(\tanh x\) and their inverses using R.
3. Compute Fisher’s confidence interval in R from two vectors x and y both:
   • directly from the formulas
   • using cor.test()

Coverage of Confidence Intervals via Simulation

1. Given a population distribution and an interval estimator that can be calculated using R, construct a single realization of the interval estimator, i.e. a confidence interval.
2. Use replicate() to repeatedly generate a random sample and calculate an associated confidence interval.
3. Check the coverage of confidence intervals returned by replicate() when given the true value of a population parameter.
4. Explain why the number of confidence intervals in a simulation that cover the true value follows a binomial distribution.
5. Use prop.test() to construct an interval estimate for the true coverage of a confidence interval using simulation.