The resource covers: Find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution Recall the meaning of the terms ‘complementary function’ and ‘particular integral’ in the context of linear differential equations, and recall that the general solution is the sum of the complementary function and a particular integral Find the complementary function for a first or second order linear differential equation with constant coefficients Recall the form of, and find, a particular integral for a first or second order linear differential equation in the cases where a polynomial or ae^bx or a cos px + b sin px is a suitable form, and in other simple cases find the appropriate coefficient(s) given a suitable form of particular integral. Use a given substitution to reduce a differential equation to a first or second order linear equation with constant coefficients or to a first order equation with separable variables. Use initial conditions to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation

Solve problems which can be modelled as the linear motion of a particle under the action of a variable force. Setting up and solving an appropriate differential equation involving variable force.

Solve problems which can be modelled as the linear motion of a particle under the action of a variable force. Setting up and solving an appropriate differential equation involving variable force.

Fit a theoretical distribution, as prescribed by a given hypothesis, to given data Use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis Use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table.

Fit a theoretical distribution, as prescribed by a given hypothesis, to given data Use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis Use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table.

Fit a theoretical distribution, as prescribed by a given hypothesis, to given data Use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis Use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table.

Use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table.

Use a probability density function which may be defined piecewise Use the general result E(g(x)) =∫f(x)g(x) dx 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.

Use a probability density function which may be defined piecewise Use the general result E(g(x)) =∫f(x)g(x) dx 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.

Use a probability density function which may be defined piecewise Use the general result E(g(x)) =∫f(x)g(x) dx 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.

Use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis

Understand the concept of a probability generating function (PGF) and construct and use the PGF for given distributions e.g 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.

Understand the concept of a probability generating function (PGF) and construct and use the PGF for given distributions e.g 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.

Understand the concept of a probability generating function (PGF) and construct and use the PGF for given distributions e.g 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.

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 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 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.

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 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 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.

Sign Test PPT Paired Sign Test PPT One Sample Wilcoxon Sign Rank Test PPT Wilcoxon-Matched-Pairs Sign-Rank Test PPT Wilcoxon Rank-Sum Test PPT

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.

Calculate a pooled estimate of a population variance from two samples Formulate hypotheses concerning the difference of population means, and apply, as appropriate – a 2-sample t-test – a test using a normal distribution

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