About this topic
Before you begin
Use sampling distributions, unbiased variance estimates and planned population-mean tests to draw contextual conclusions in H2 Mathematics.
Learning goals
- Reason about sampling and sampling distributions
- Calculate a population-mean estimate from a specified independent sample
- Calculate an unbiased population-variance estimate from a specified independent sample
- Calculate an estimated standard error from a specified independent sample
- Calculate a sample mean's expectation from specified population moments
- Calculate a sample mean's variance from specified population moments
- Calculate a sample mean's standard error from specified population moments
- Calculate a sample-mean probability under a specified exact or approximate normal model
- Calculate a minimum sample size for specified normal-model precision
- Formulate and interpret a justified hypothesis test
- Calculate a signed population-mean test statistic under a stipulated normal-test model
- Calculate a population-mean test p-value for a prespecified alternative
- Calculate all standardised critical boundaries for a planned population-mean test
- Calculate all sample-mean critical boundaries for a planned known-variance normal test
- Determine a population-mean test decision using the unrounded planned p-value and level
Syllabus statements covered
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
Choose a lesson
Work through them in order.
- Not startedSampling means and unbiased varianceNextChoose sampling distributions, estimate population variance from raw or coded data, and plan precision without confusing bias with spread.
- Not startedPopulation-mean hypothesis testsNextCarry out planned one- and two-tailed mean tests, distinguish exact and estimated-spread models, and interpret the evidence in context.