11-12.2 Continuous random variables and the normal distribution
11-12. General continuous random variables:
11-12.ACMMM164 use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
11-12.ACMMM165 understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts
11-12.ACMMM166 recognise the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases
11-12.ACMMM167 understand the effects of linear changes of scale and origin on the mean and the standard deviation.
11-12. Normal distributions:
11-12.ACMMM168 identify contexts such as naturally occurring variation that are suitable for modelling by normal random variables
11-12.ACMMM169 recognise features of the graph of the probability density function of the normal distribution with mean μ and standard deviation ∑ and the use of the standard normal distribution
11-12.ACMMM170 calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems.
11-12.ACMMM173 use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli.
11-12. Sample proportions:
11-12.ACMMM174 understand the concept of the sample proportion p as a random variable whose value varies between samples, and the formulas for the mean p and standard deviation √(p(1–p)/n of the sample proportion p
11-12.ACMMM175 examine the approximate normality of the distribution of p for large samples
11-12.ACMMM176 simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of p and the approximate standard normality of p - p /√(p(1–p)/n where the closeness of the approximation depends on both n and p.
11-12. Confidence intervals for proportions:
11-12.ACMMM177 the concept of an interval estimate for a parameter associated with a random variable
11-12.ACMMM178 use the approximate confidence interval p–z√(p(1–p)/n, p+z√(p(1–p)/n, as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution
11-12.ACMMM179 define the approximate margin of error E = z√(p(1–p)/n and understand the trade-off between margin of error and level of confidence
11-12.ACMMM180 use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain p.
11-12.3 Introduction to differential calculus
11-12. Rates of change:
11-12.ACMMM077 interpret the difference quotient f(x+h)–f(x)h as the average rate of change of a function f
11-12.ACMMM078 use the Leibniz notation δx and δy for changes or increments in the variables x and y
11-12.ACMMM079 use the notation δy/δx for the difference quotient f(x+h)–f(x)h where y = f(x)
11-12.ACMMM080 interpret the ratios f(x+h)–f(x)/h and δy/δx as the slope or gradient of a chord or secant of the graph of y = f(x).
11-12. The concept of the derivative:
11-12.ACMMM081 examine the behaviour of the difference quotient f(x+h)–f(x)/h as h → 0 as an informal introduction to the concept of a limit
11-12.ACMMM082 define the derivative f'(x) as limh→0 f(x+h)–f(x)/h
11-12.ACMMM083 use the Leibniz notation for the derivative: dy/dx = limδx→0 δy/δx and the correspondence dy/dx = f'(x) where y = f(x)
11-12.ACMMM084 interpret the derivative as the instantaneous rate of change
11-12.ACMMM085 interpret the derivative as the slope or gradient of a tangent line of the graph of y = f(x).
11-12. Computation of derivatives:
11-12.ACMMM086 estimate numerically the value of a derivative, for simple power functions
11-12.ACMMM087 examine examples of variable rates of change of non-linear functions
11-12.ACMMM088 establish the formula d/dx(xn) = nx to the (n–1) power for positive integers n by expanding (x + h)n or by factorising (x + h)n -xn.
11-12. Properties of derivatives:
11-12.ACMMM089 understand the concept of the derivative as a function
11-12.ACMMM090 recognise and use linearity properties of the derivative
11-12.ACMMM091 calculate derivatives of polynomials and other linear combinations of power functions.
11-12. Applications of derivatives:
11-12.ACMMM092 find instantaneous rates of change
11-12.ACMMM093 find the slope of a tangent and the equation of the tangent
11-12.ACMMM094 construct and interpret position-time graphs, with velocity as the slope of the tangent
11-12.ACMMM095 sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as x → ∞ and x → –∞
11-12.ACMMM096 solve optimisation problems arising in a variety of contexts involving simple polynomials on finite interval domains.
11-12.ACMMM097 calculate anti-derivatives of polynomial functions and apply to solving simple problems involving motion in a straight line.
11-12.3 Discrete random variables
11-12. General discrete random variables:
11-12.ACMMM136 understand the concepts of a discrete random variable and its associated probability function, and their use in modelling data
11-12.ACMMM143 use a Bernoulli random variable as a model for two-outcome situations
11-12.ACMMM144 identify contexts suitable for modelling by Bernoulli random variables
11-12.ACMMM145 recognise the mean p and variance p(1–p) of the Bernoulli distribution with parameter p
11-12.ACMMM146 use Bernoulli random variables and associated probabilities to model data and solve practical problems.
11-12. Binomial distributions:
11-12.ACMMM147 understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of 'successes' in n independent Bernoulli trials, with the same probability of success p in each trial
11-12.ACMMM149 determine and use the probabilities P(X = r) = (n r)p to the r power (1–p) to the (n–r) power associated with the binomial distribution with parameters n and p; note the mean np and variance np(1–p) of a binomial distribution
11-12.ACMMM050 use set language and notation for events, including A(or A') for the complement of an event A for the A∩B intersection of events and A and B, and A∪B for the union, and recognise mutually exclusive events