2 interactive labs · AS & A Level Mathematics 9709
Sample Means and Estimates
6.4 Sampling and estimation · Draw many seeded random samples, watch the sample means centre on μ with variance σ² ÷ n and turn normal, and compare s² with divisors n and n − 1.
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About this lab
Draw many seeded random samples, watch the sample means centre on μ with variance σ² ÷ n and turn normal, and compare s² with divisors n and n − 1.
You will be able to
State and use E(X̄) = μ and Var(X̄) = σ²/n, and check them against the mean and variance of many sample means drawn from the same population.
Explain that X̄ is exactly normal for a normal population and approximately normal for a large sample from any population (informal Central Limit Theorem), and use X̄ ~ N(μ, σ²/n) to find a probability.
Calculate unbiased estimates of the population mean and variance from a sample, s² = (Σx² − (Σx)²/n) ÷ (n − 1), and explain why dividing by n gives a value that is too small on average.
Confidence Intervals
6.4 Sampling and estimation · Build an interval for a mean or a proportion from the MF19 z value, change n, σ and the level, then draw 100 samples to see what “95% confident” means.
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About this lab
Build an interval for a mean or a proportion from the MF19 z value, change n, σ and the level, then draw 100 samples to see what “95% confident” means.
You will be able to
Calculate a confidence interval for a population mean (normal population with known variance, or a large sample) using z from the critical values table.
Calculate an approximate confidence interval for a population proportion from a large sample.
Explain how the width of an interval changes with n, σ and the confidence level, and interpret it correctly.
