# Archive | Notes

## Hypothesis Testing for the Variance or Standard Deviation

The sampling distribution for the variance of a normal distribution with varianceis given by This means that if a random sample of sizeis taken from a normally distributed population with variancethen the random variablehas thedistribution withdegrees of freedom. To perform a hypothesis test for the hypothesis Is the population normal? State the null hypothesis – […]

## Confidence Interval For The Mean of a Normal Distribution When the Standard Deviation is Not Given

If samples of sizeare taken from a population whose meanand standard deviationis known thenthe mean of the sample has the normal distributionIf we know the standard deviation but not the mean of the population then we can find a confidence interval for the mean of the population by rearranging(1) to give (2) with probability corresponding […]

## Unbiased Estimators

An estimator for a statistical parameteris said to be unbiased if The mean of a sampleis an unbiased estimator of the population meanThis can be used in a less obvious way in the following example. A bag has 500 beans of two colours, black and white. A sample of twenty beads is randomly selected, with […]

## The Central Limit Theorem

The central limit theorem states: Ifis a random sample of sizedrawn from any population with meanand variancethen the sample meanhas expected valueand expected variance If lots of samples, all of sizeare taken from the population, then the distribution of the sample means is approximately normally distributed,and the goodness of the fit improves with increasingThe underlying […]

## Testing for the Equality of Two Means When the Sample Sizes are Large

To test for the equality of the mean of a population, we can use the central limit theorem, which states that the mean of a sample of size n from any population is approximately normally distributed, with the accuracy of the approximation improving with increasing n. When the variance of the population,is not known and […]

## Primary and Secondary Data

Primary data is data collected by a person or organisation as a result of a survey, census or test of some sort. Collection primary data can be time consuming and expensive, but it has several important advantages. The collection method can be tailored to the purpose. The accuracy of the data is easier to assess. […]

## Differentiuating Between Unbiased Estimators

When choosing an estimator for a statisitical parameter we are primarily interested in simplicity of calculation and the bias of the estimator. The bias of a statisitical parameter is the difference between the estimator for the parameter and the true value of the parameter. If the estimator for a population parameteristhen We are usually interested […]

## Biased Estimators

An estimatorfor a statistical parameteris said to be biased if Bias is often impossible to avoid in practice and must be taken into account when statisical calculations are performed. Example: To estimate the number,of nesting birds, scientists catch 100, tag them and release them. The fraction of birds with tags isLater, they catch another 100 […]

## Systematic Sampling

Systematic sampling is a method of taking a ‘random’ sample. The population has to listed and ordered in some way, then ever nth element of the list is chosen. In order that any of the first n in the list can be chosen, the first to be chosen may be done randomly – often with […]

## Stratified Sampling

Stratified sampling is a sampling method useful when the population to be sampled falls into distinct categories. It is desirable that the sample reflect the population, so that the proportion of each category in the sample reflects the proportion of that category in the population. Suppose that school pupils are to be surveyed on their […]