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A-Level Specifications
Textbook
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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 real-world 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

Chi-Squared 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 chi-squared (X^2) family of distributions

  • Be able to test a hypothesis

  • Apply goodness-of-fit tests to discrete data

  • Use contingency tables

  • Apply goodness-of-fit 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 F-distribution

  • Carry out na F-test to test whether two independent random variables are from normal distributions with equal variances

Confidence Intervals and Tests using the                t-distribution

  • 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 t-test

  • 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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