Topic 4.3 Statistical Inference

Syllabus

Sample means

4.3.1 examine the concept of the sample mean \bar X as a random variable whose value varies between samples where X is a random variable with mean \mu and the standard deviation \sigma

4.3.2 simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of \bar X across samples of a fixed size n, including its mean \mu its standard deviation \frac{\sigma}{\sqrt n} (where \mu and \sigma are the mean and standard deviation of X), and its approximate normality if n is large

4.3.3 simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate the approximate standard normality of \frac{\bar X-\mu}{s\big/\sqrt n} for large samples ( n\geq 30) , where s is the sample standard deviation

Confidence intervals for means

4.3.4 examine the concept of an interval estimate for a parameter associated with a random variable

4.3.5 examine the approximate confidence interval \Big( \bar X-\frac{zs}{\sqrt n},\bar X+\frac{zs}{\sqrt n} \Big) as an interval estimate for the population mean \mu, where z is the appropriate quantile for the standard normal distribution

4.3.6 use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain \mu

4.3.7 use \bar x and s to estimate \mu and \sigma to obtain approximate intervals covering desired proportions of values of a normal random variable, and compare with an approximate confidence interval for \mu

Lessons

1. Sample mean as a random variable

2. Distribution of sample means

Khan academy https://www.khanacademy.org/math/probability/statistics-inferential/sampling-distribution/v/central-limit-theorem. Work through the sequence of five videos. The onlinestatbook simulator referenced can be found at http://onlinestatbook.com/stat_sim/sampling_dist/.

3. Standardised sample means

We can use the sample standard deviation as an estimator of population standard deviation and this allows us to model the sampling distribution of the mean as a normal distribution with standard deviation equal to the sample standard deviation, and unknown mean. The whole point of this is to allow us to draw inferences about the population mean.

See the simulator at http://maths.prideaux.id.au/Shiny/standardNormalSim/ (this is experimental … if it works it should show you how the means of samples can be used to approximate a normal distribution using \frac{\bar X-\mu}{s\big/\sqrt n} approximating a standard normal distribution. Note that the private server this simulator sits on may not run all the time … I make sure it is up when we are working on this content but it may be down at other times.)

4. Interval estimates

5. Confidence interval for the mean of a random variable

Khan academy https://www.khanacademy.org/math/probability/statistics-inferential/confidence-intervals/v/confidence-interval-1. Work through the sequence of three videos. EDIT: These videos are about confidence interval for population proportion, and so more relevant to the Methods course than the Specialist course.

6. Estimating population mean and standard deviation

Other resources

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