# 23 Statistics 4

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Statistics 1 and Statistics 2 and Core 1, Core 2 and Core 3.

Those candidates who have not studied the module Statistics 3 will also require knowledge of the mean, variance and standard deviation of a difference between two independent normal random variables.

Candidates 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 an exponential distribution, $\mbox{F}(x) = 1-\mathrm{e}^{- \lambda x}$

Efficiency of $\space Estimator A\space$ relative to $\space \displaystyle Estimator B = \frac{1/\text{Var}(Estimator A)}{1/\text{Var}(Estimator B)}$

 Conditions for application of a geometric distribution. Calculation of probabilities for a geometric distribution using formula. Mean, variance and standard deviation of a geometric distribution. Knowledge and derivations will be expected. Conditions for application of an exponential distribution. Knowledge that lengths of intervals between Poisson events have an exponential distribution. Calculation of probabilities for an exponential distribution. Using cumulative distribution function or integration of probability density function. Mean, variance and standard deviation of an exponential distribution. Knowledge and derivations will be expected.

 Review of the concepts of a sample statistic and its sampling distribution, and of a population parameter. Estimators and estimates. Properties of estimators. Unbiasedness, consistency, relative efficiency. Mean and variance of pooled estimators of means and proportions. Proof that $\space \mbox{E}(S^2) = \sigma^2$.
 Confidence intervals for the difference between the means of two normal distributions with unknown variances. Independent and paired samples. For independent samples, only when the population variances may be assumed equal so that a pooled estimate of variance may be calculated. Small samples only. Using a $\space t$ distribution. Confidence intervals for a normal population variance (or standard deviation) based on a random sample. Using a $\space \chi^2$ distribution. Confidence intervals for theratio of two normal population variances (or standard deviations) based on independent random samples. Introduction to  $F$ distribution. To include use of the supplied tables. Using an  $F$ distribution.
 The significance level to be used in a hypothesis test will usually be given. Tests for the difference between the means of two normal distributions with unknown variances. Independent and paired samples. For independent samples, only when the population variances may be assumed equal so that a pooled estimate of variance may be calculated. Small samples only. Using a  $t$-statistic. Tests for a normal population variance (or standard deviation) based on a random sample. Using a  $\chi^2$-statistic. Tests for the ratio of two normal population variances (or standard deviations) based on independent random samples. Using a  $F$-statistic.
 Use of  $\sum \frac{(O_i - E_i)^2}{E_i}$ as an approximate  $\chi^2$-statistic. Conditions for approximation to be valid. The convention that all  $E_i$ should be greater than 5 will be expected. Goodness of fit tests. Discrete probabilities based on either a discrete or a continuous distribution. Questions may be set based on a knowledge of discrete or continuous random variables from the module Statistics 2. Integration may be required for continuous random variables.