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 […]

## Hypothesis Testing for the Equality of the Means of Two Populations if the Common Variance is Not known

When independent samples of sizeandare taken from two normally distributed populations with means and and known population standard deviationsandthe random variableis normally distributed with meanand variance is normally distributed. We can test for the equality ofandby doing a hypothesis test usingIfthen (1) Ifis not known then we cannot use the last expression above. We can […]

## An Example of Bias

Suppose that a random variableis known to be uniform so thatA sample of size n is taken and the maximumis recorded so that Given any number 0<x<a, the probability thatis The probability thatandis The probability thatandis Continuing in this way, for the sample of sizethe probability that all theare less thanisso that The probability density […]

## Proof That the Pooled Standard Deviation is an Unbiased Estimator For the Standard Deviation if The Population Variances are Equal

The pooled standard deviation is found by taking a weighted average of the variances of two samples,andand finding the square root of the result. The equation is We can easily show thatis an unbiased estimator for if sinceandare both unbiased estimators for so that and

## Constructing Confidence Intervals for the Difference of Two Means From Two Normal Populations

If we have two populations with population statisticsandthen the mean difference between the means of two samples isand if the sample sizes areand (from populations 1 and 2 respectively) are large, then difference between the sample means is normally distributed:(from the central limit theorem). When the sample sizes are small we need to make the […]

## Hypothesis Testing for the Difference of Two Means From Two Normal Populations

If we have two populations with population statisticsandthen the mean difference between the means of two samples isand if the sample sizes areand (from populations 1 and 2 respectively) are large, then difference between the sample means is normally distributed:(from the central limit theorem). When the sample sizes are small we need to make the […]

## Consistent Estimators

Given a sample from a population, we can estimate the mean of the population fromwidth=”75″ height=”45″ hspace=”8″> As the sample size increases, the variance ofdecreases. This property makesa useful estimator for the population mean,since by increasing the sample size n, we can reduce the variance ofIfis also an unbiased estimator for the population meanthenis a […]

## The Geometric Distribution

The geometric distribution models players of a game ‘in search of success’ . When a player wins they stop playing. There are three conditions the game must satisfy: When the player wins he stops playing, or at least the geometric distribution ceases to model the game at this point. If the player continues to play, […]

## Summary of Moment Generating Functions

The moment generating function for a discrete distribution is defined asfor a continuous distribution asand the moment generating function of a random variableisThey are called moment generating functions because we can obtain the moments of a distribution from them. Definition Themoment of a discrete random variable with probability mass functionand is The first moment is […]