4. Further Probability & Statistics

• use a probability density function which may be defined piecewise

• use the general result X xx xE g fg d=_ ^ ^^hi hhy where f(x) is the probability density function of the continuous random variable X and g(X) is a function of X

• understand and use the relationship between the probability density function (PDF) and the cumulative distribution function (CDF), and use either to evaluate probabilities or percentiles

• use cumulative distribution functions (CDFs) of related variables in simple cases. e.g. given the CDF of a variable X, find the CDF of a related variable Y, and hence its PDF, e.g. where Y = X 3.

• formulate hypotheses and apply a hypothesis test concerning the population mean using a small sample drawn from a normal population of unknown variance, using a t-test

• calculate a pooled estimate of a population variance from two samples Calculations based on either raw or summarised data may be required.

• formulate hypotheses concerning the difference of population means, and apply, as appropriate - a 2-sample t-test - a paired sample t-test - a test using a normal distribution The ability to select the test appropriate to the circumstances of a problem is expected.

• determine a confidence interval for a population mean, based on a small sample from a normal population with unknown variance, using a t-distribution

• determine a confidence interval for a difference of population means, using a t-distribution or a normal distribution, as appropriate.

4.3.1Fitted distributions

• fit a theoretical distribution, as prescribed by a given hypothesis, to given data Questions will not involve lengthy calculations.

4.3.2Goodness-of-fit test

• use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis Classes should be combined so that each expected frequency is at least 5.

4.3.3Independence test

• use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table. Yates' correction is not required. Where appropriate, either rows or columns should be combined so that the expected frequency in each cell is at least 5.

• understand the idea of a non-parametric test and appreciate situations in which such a test might be useful e.g. when sampling from a population which cannot be assumed to be normally distributed.

• understand the basis of the sign test, the Wilcoxon signed-rank test and the Wilcoxon rank-sum test Including knowledge that Wilcoxon tests are valid only for symmetrical distributions.

• use a single-sample sign test and a single- sample Wilcoxon signed-rank test to test a hypothesis concerning a population median Including the use of normal approximations where appropriate. Questions will not involve tied ranks or observations equal to the population median value being tested.

• use a paired-sample sign test, a Wilcoxon matched-pairs signed-rank test and a Wilcoxon rank-sum test, as appropriate, to test for identity of populations. Including the use of normal approximations where appropriate. Questions will not involve tied ranks or zero-difference pairs.

• understand the concept of a probability generating function (PGF) and construct and use the PGF for given distributions Including the discrete uniform, binomial, geometric and Poisson distributions.

• use formulae for the mean and variance of a discrete random variable in terms of its PGF, and use these formulae to calculate the mean and variance of a given probability distribution

• use the result that the PGF of the sum of independent random variables is the product of the PGFs of those random variables.