Sampling and inference
Connect sampling design, mean uncertainty and unbiased variance estimates to planned population-mean tests and contextual conclusions.
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Before you begin
Connect sampling design, mean uncertainty and unbiased variance estimates to planned population-mean tests and contextual conclusions.
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.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Sampling means and unbiased varianceChoose sampling distributions, estimate population variance from raw or coded data, and plan precision without confusing bias with spread.
- Population-mean hypothesis testsCarry out planned one- and two-tailed mean tests, distinguish exact and estimated-spread models, and interpret the evidence in context.