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.

In class we will view a simulator that shows how the means of samples can be used to approximate a normal distribution using $z=\frac{\bar X-\mu}{s\big/\sqrt n}$ approximating a standard normal distribution. (People with the required technical skills to install and use R-Shiny may be interested in running it for themselves. If that’s you, the source for the simulator will be provided on request.)

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

HyperStat Online. Chapters 6 and 8 are relevant. (Other parts may be relevant for the work in Mathematics Methods.)
Sampling Distribution of the Sample Mean from the Wolfram Demonstrations Project by Jim R Larkin

Topic 4.3 Statistical Inference

Syllabus

Sample means

Confidence intervals for means

Lessons

1. Sample mean as a random variable

2. Distribution of sample means

3. Standardised sample means

4. Interval estimates

5. Confidence interval for the mean of a random variable

6. Estimating population mean and standard deviation

Other resources

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Syllabus

Sample means

Confidence intervals for means

Lessons

1. Sample mean as a random variable

2. Distribution of sample means

3. Standardised sample means

4. Interval estimates

5. Confidence interval for the mean of a random variable

6. Estimating population mean and standard deviation

Other resources

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