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Topic 5: Statistical inference

Sample means

7 hours

• Understand the concept of the sample mean X̄ as a random variable whose value varies between samples where X is a random variable with mean μ and the standard deviation σ. 1 interactive
• Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine properties of the distribution of X̄ across samples of a fixed size n, including its mean μ, its standard deviation σ/√n (where μ and σ are the mean and standard deviation of X) and its approximate normality if n is large.
• Recognise and use the link between the normal distribution of the sample mean and the statistical notation X̄ ~ N(μ, σ²/n).
• Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine the approximate standard normality of (X̄ − μ)/(s/√n) for large samples (n ≥ 30), where s is the sample standard deviation (Central limit theorem). 1 interactive
• Model and solve problems that involve sample means, with and without technology.

Confidence intervals for means

6 hours

• Understand the concept of an interval estimate for a parameter associated with a random variable.
• Understand and use the approximate confidence interval (x̄ − z s/√n, x̄ + z s/√n), as an interval estimate for μ, the population mean, where z is the appropriate quantile for the standard normal distribution. 1 interactive
• Understand and use the approximate margin of error.
• Understand and use the relationship between margin of error, level of confidence and sample size.
• Understand and use the concept that there are variations in confidence intervals between samples and that most but not all confidence intervals contain μ. 1 interactive
• Use x̄ and s to estimate μ and σ, to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for μ. 1 interactive
• Model and solve problems that involve interval estimates for sample means, with and without technology.