# Author Archive | A*Maths

## The Mann Whitney U Test

The Mann Whitney U test allows the comparison of two independent random samples (1 and 2). The Mann Whitney U statistic is defined as: – where samples of sizeandare pooled and Ri are the ranks of sample 1. U can be resolved as the number of times observations in one sample precede observations in the […]

## Using the F – Distribution to Test for the Equality of the Variances of Two Normal Populations

If two populations are normally distributed with distributionsandwe can test the hypothesis thatusing the F – distribution. We take a sample from populations 1 and 2, of sizesandrespectively, and find the sample variances,andrespectively. The test statistic is thenand assuming thatwill follow adistribution withandWe can find the critical value corresponding to the significance level of the […]

## Pooled Standard Deviation

Larger samples are more use than smaller samples. The hypothesis tests conducted using larger samples are more reliable and accurate. Often though, it is impossible or too expensive to carry out a large survey. An alternative is to group smaller similar surveys together. The meanof a number of surveys, each of sizeand meanis easily found: […]

## The Two Sample t – Test

If we have two independent populations with unknown variances, then we can use the t – distribution to test hypotheses about the means of the two populations – either that they are equal or that they differ by a certain amount. The procedure is: Write down the null hypothesisfor exampleor Write down the alternative hypothesisfor […]

## Degrees of Freedom

The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. A simple example is shown in calculating the mean of a sample. Supposeis a sample from a population. Let Theare estimates of the residualswhereis the true mean of the population. The sum […]

## Bias

If a test statisticis used as an estimate for a population parameterthen in general we will not expect the value ofto equal– the estimator may be biased forThe bias of the estimator is given by An estimatorfor a population parameteris an unbiased estimator forif An unbiased estimator is always better than a biased estimator, and […]

## The Exponential Distribution

The probability distribution for the exponential function, with probability distribution functionis illustrated below. A number of situations can be modelled by the exponential distribution. In these examples it must be assumed however that each event is independent. The lifetimes of electric bulbs. Some bulbs will fail almost immediately and some will last a very long […]

## Factors Affecting the Power of a Hypothesis Test

The power of a hypothesis test is the probability of not committing a Type II error – failing to reject the null hypothesis when the null hypothesis is false. The effect size is the difference between the true value and the value specified in the null hypothesis. Effect size = True value – Hypothesized value […]

## Confidence Interval for the Variance of a Normal Distribution

In practice, though a population may have a ‘true’ value for the variance, this is never know and the variance is always estimated from a sample using the formulaWe can use this to find a confidence interval for the unknown varianceof whichis an estimate. We can do this using the fact thatthedistribution with degrees of […]

## Power Function of a Test

When hypothesis testing, the power of a test is the probability of not committing a Type II error. A Type II error is committed if a false null hypothesis is not rejected. We may think of the power of a hypothesis test as the ability of the test to reject a false null hypothesis. It […]