22 Statistics 3

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Statistics 1 and 2 and Core 1 and 2.

Candidate may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

For $\space X_i$ independently distributed $\space (\mu_i, \space \sigma_i^2)$, then

$\Sigma a_i X_i \space$ is distributed$\space \big(\Sigma a_i \mu_i, \space \Sigma a_i^2 \sigma_i^2 \big)$

Power = 1 – P(Type $\space \mathrm{II}$ error)

 Bayes’ Theorem. Knowledge and application to at most three events. The construction and use of tree diagrams
 Mean, variance and standard deviation of a linear combination of two (discrete or continuous) random variables. To include covariance and correlation. Implications of independence. Applications, rather than proofs, will be required. Mean, variance and standard deviation of a linear combination of independent (discrete or continuous) random variables. Use of these, rather than proofs, will be required. Linear combinations of independent normal random variables. Use of these only.
 Mean, variance and standard deviation of binomial and Poisson distributions. Proofs using  $\mbox{E}(X)$  and  $\mathrm{E}(X(X-1))$ together with  $\sum p_i = 1 \space$. A Poisson distribution as an approximation to a binomial distribution. Conditions for use. A normal distribution as an approximation to a binomial distribution. Conditions for use. Knowledge and use of continuity corrections. A normal distribution as an approximation to a Poisson distribution. Conditions for use. Knowledge and use of continuity corrections.
 Estimation of sample sizes necessary to achieve confidence intervals of a required width with a given level of confidence. Questions may be set based on a knowledge of confidence intervals from the module Statistics 1. Confidence intervals for the difference between the means of two independent normal distributions with known variances. Symmetric intervals only. Using a normal distribution. Confidence intervals for the difference between the means of two independent distributions using normal approximations. Large samples only. Known and unknown variances. The mean, variance and standard deviation of a sample proportion. Unbiased estimator of a population proportion. $\hat{P}$ A normal distribution as an approximation to the sampling distribution of a sample proportion based on a large sample. $\displaystyle\mbox{N}\Bigg(p, \frac{p(1-p)}{n} \Bigg)$ Approximate confidence intervals for a population proportion and for the mean of a Poisson distribution. Using normal approximations. The use of a continuity correction will not be required in these cases. Approximate confidence intervals for the difference between two population proportions and for the difference between the means of two Poisson distributions. Using normal approximations. The use of continuity corrections will not be required in these cases.
 The notion of the power of a test. Candidates may be asked to calculate the probability of a Type $\space\mathrm{II}$ error or the power for a simple alternative hypothesis of a specific test, but they will not be asked to derive a power function. Questions may be set which require the calculation of a $\space z$-statistic using knowledge from the module Statistics 1. The significance level to be used in a hypothesis test will usually be given. Tests for the difference between the means of two independent normal distributions with known variances. Using a  $z$-statistic. Tests for the difference between the means of two independent distributions using normal approximations. Large samples only. Known and unknown variances. Tests for a population proportion and for the mean of a Poisson distribution. Using exact probabilities or, when appropriate, normal approximations where a continuity correction will not be required. Tests for the difference between two population proportions and for the difference between the means of two Poisson distributions. Using normal approximations where continuity corrections will not be required. In cases where the null hypothesis is testing an equality, a pooling of variances will be expected. Use of the supplied tables to test $\mbox{H}_0 :\rho = 0 \space$ for a bivariate normal population. Where $\space \rho$ denotes the population product moment correlation coefficient.