Inference with the normal and t-distributions

Learning objectives

State that the sample mean of a random sample of size n has mean μ and variance σ²/n, and apply the central limit theorem to justify treating the sample mean as approximately normal for a large sample from any population

Calculate an unbiased estimate of the population variance from sample data using s² = Σ(x − x̄)²/(n − 1), and explain why dividing by n − 1 rather than n removes the bias

Construct a confidence interval for a population mean, using the normal distribution when σ is known or the sample is large, and the t-distribution with n − 1 degrees of freedom when the sample is small and drawn from a normal population with unknown variance, and construct a confidence interval for the difference of two population means using the normal or the t-distribution as appropriate

Carry out a hypothesis test for a population mean by stating the null and alternative hypotheses, computing the test statistic, comparing it with the critical value or comparing the p-value with the significance level, and writing a conclusion in the context of the question

Carry out a two-sample test for a difference of means, using a normal distribution when the population variances are known or both samples are large, and pooling the two sample variances and using the t-distribution with n₁ + n₂ − 2 degrees of freedom when the populations are normal with a common unknown variance, and use a paired-sample t-test on the differences when the data are naturally paired

Define a Type I error as rejecting a true null hypothesis and a Type II error as failing to reject a false one, and state that the probability of a Type I error equals the significance level of the test