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MATHEMATICS TEACHER
Mr Valmonte's Maths Page
ALevel Specifications
Textbook
Further Statistics 1 OBJECTIVES
Discrete Random Variables

Find the expected value of a discrete random variable X

Find the expected value of X^2

Find the variance of a discrete random variable

Use the expected value and variance of a function X

Solve problems involving random variables
Poisson Distribution

Use the Poisson distribution to model realworld situations

Use the additive property of the Poisson distribution

Understand and use the mean and variance of the Poisson distribution

Understand and use the mean and variance of the binomial distribution

Use the Poisson distribution as an approximation to the binomial distribution
Geometric and Negative Binomial Distribution

Understand and use the geometric distribution

Calculate and use the mean and variance of the geometric distribution

Understand and use the negative binomial distribution

Calculate and use the mean and variance of the negative binomial distribution
Hypothesis Testing

Use hypothesis tests to test for the mean of a Poisson distribution

Find critical regions of a Poisson distribution using tables

Use hypothesis tests to test for the parameter p in a geometric distribution

Find critical regions of a geometric distribution
Central Limit Theorem

Understand and apply the central limit theorem to approximate the sample mean of a random variable, X

Apply the central limit theorem to other distributions
ChiSquared Tests

Form hypothesis about how well a distribution fits as a model for an observed frequency distribution and measure goodness of fit of a model to observed data

Understand degrees of freedom and use the chisquared (X^2) family of distributions

Be able to test a hypothesis

Apply goodnessoffit tests to discrete data

Use contingency tables

Apply goodnessoffit tests to geometric distributions
Probability Generating Functions

Understand the use of probability generating functions

Use probability generating functions for standard distributions

Use probability generating functions to find the mean and variance distribution

Know the probability generating function of the sum of independent random variables
Quality of Tests

know about Type I and Type II errors

Find Type I and Type II errors using the normal distribution

Calculate the size and power of a test

Draw a graph of the power function for a test
Further Statistics 2 OBJECTIVES
Linear Regression

Calculate the equation of a regression line using raw data or summary statistics

Use coding to find the equation of a regression line

Calculate residuals and use them to test for linear fit and identify outliers

Calculate re residual sum of squares (RSS)
Correlation

Calculate the value of the product moment correlation coefficient

Understand the effect of coding on it and understand the conditions for its use

Calculate and interpret Spearman's rank correlation coefficient

Carry out hypothesis tests for zero correlation using either Spearman correlation coefficient rank or the product moment correlation coefficient
Continuous Distributions

Understand and use the probability density function for a continuous random variable

Understand and use the cumulative distribution function for continuous random variable

Find the mean, variance, mode, median and percentiles of a continuous random variable and describe the skewness

Understand, use and model situations using the continuous uniform distribution
Combinations of Random Variables

Find the distribution of linear combinations of normal random variables

Solve modelling problems involving combinations of normal random variables
Estimation, Confidence Intervals and Tests using a Normal Distribution

Understand and use estimates and estimators

Undestand bias

Find the standard error

Calculate and use confidence intervals for population parameters

Carry out hypothesis tests for the difference between the means of two normally distributed random variables with known variances

Carry out hypothesis tests using large sample results in cases where the population variance is unknown
Further Hypothesis Tests

Find a confidence interval for the variance of a normal distribution

Conduct a hypothesis test for the variance of a normal distribution

Understand and use the Fdistribution

Carry out na Ftest to test whether two independent random variables are from normal distributions with equal variances
Confidence Intervals and Tests using the tdistribution

Find a confidence interval for the mean of a normal with unknown variance

Conduct a hypothesis test for the mean of a normal distribution with unknown variance

Carry out a paired ttest

Find a confidence interval for the difference between two means from independent normal distributions with equal but unknown variances

Conduct a hypothesis test for the difference between two means from independent normal distributions with equal but unknown variances
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