# Archive | S2

## Hypothesis Testing – Critical Values and Regions

There is a certain method of conducting hypothesis tests which makes it very suitable for automation. If we could know the set of values of a statistic – typically a single occurrence or mean – which results in the null hypothesis being rejected, these could be written down on the side of a machine and […]

## Probability Distribution For the Value of Coins When Coins Are Taken From One Bag and Put Into Another

Suppose we have two bags of coins, both with 1p and 2p pieces. One coin is taken from bag 1 and put into bag 2. Two coins are then taken from bag 2. I want to find the probability distribution for the value of coins taken from bag 2. The probability that a 2p coin […]

## The Coefficient of Determination

Pearson’s correlation coefficient is given by Pearson’s correlation coefficient r tells us how good a fit to a straight line a set of data points is. If the data is a perfect fit to a straight line, and ifthere is absolutely no fit that can be made to a straight line. Almost as useful is […]

## Populations, Samples and Censuses

A population is the set of individuals in a group. Often information needs to be collected about the population, Taking a census and taking a sample are methods of collecting data about the population. A govenrnent for example needs to plan the supply of public services in advance. A hospital might cost £100m to build, […]

## The Cumulative Distribution Function

A probability distribution is usually defined in terms of it’s probability distribution function (if continuous), the probability that it takes a value in a certain range, or probability mass function (if discrete), the probability that it takes a certain value Sometimes it is more convenient to define it in terms of it cumulative distribution function. […]

## Mode of a Continuous Distribution

The mode of a continuous distribution is the most probable value of the random variable associated with the distribution. If we sketch the probability distribution function againstit will typically have a maximum. Finding the mode then becomes a matter of finding the turning point, or solving the equation Example: Find the mean of the probability […]

## Formulating Null and Alternative Hypotheses

Every hypothesis test seeks to prove or disprove a statistical statement about a population to a particular level of certainty. In order to do this the statement needs to be clearly phrased. Typically the statement is about the mean of a distribution or the probability of an event occurring. The null hypothesis is the value […]

## Prediction and Extrapolation

Extrapolation is a bad idea. It involves using a set of data to make predictions for data outside the range used to construct a statistical model. Suppose for example that a drug is being tested. The drug is designed to lower cholesterol. The drug is tested on healthy people first to minimise the risks of […]

## Normal Approximation to The Poisson Distribution and the Continuity Correction

The Poisson distribution, writtenhas Expectation ValueThis is the expected number of successes in n attempts. The variance is given byIf we want to use the normal distribution as an approximation to estimatefor example – which is very useful whenandare large – we must make modifications since the Poisson distribution is a discrete distribution but the […]

## Normal Approximation to The Binomial Distribution and the Continuity Correction

The binomial distribution, writtenhas Expectation ValueThis is the expected number of successes in n attempts. The variance is given by If we want to use the normal distribution as an approximation to estimatefor example – which is very useful when n is large – we must make modifications since the binomial distribution is a discrete […]