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Achievement Standard Statistics and Modelling 3.1 Achievement: Determine the trend for time series data 3 Credits Internal I can Nulake Sigma graph time series. 2 - 44 Ch 13 smooth a time series with moving averages. 20-28 Ch 13 describe the trend in context using smoothed data. 29-33 Ch 13 calculate and/or estimate seasonal effects. 34-44 Ch 13 make forecasts based on a trend line fitted to smoothed data with seasonal effects. 34-44 Ch 13 Merit: I can I can analyse a time series based on data that cannot be modelled by a single straight linear trend over its entire range. choose and justify a model for the analysis of a time series Excellence seasonally adjust data and interpret the results in context compare two sets of related time series data use formal methods of analysis (eg least squares regression lines). report on the relevance and usefulness of forecast discuss the features of the time series data report on potential sources of bias suggest improvements to the model used for the analysis of a time series comment on limitations of the analysis of a time series Time series data may be represented by an index series. Use of appropriate technology is expected. Achievement Standard Statistics and Modelling 3.2 Merit: Achievement: Calculate confidence intervals for population parameters 3 Credits I can Nulake Sigma calculate large sample confidence intervals for the mean of a population 80-83 Ex14.1 calculate large sample confidence intervals for the proportion of a population 84-87 Ex14.3 I can Nulake Sigma interpret confidence intervals 97-101 calculate the sample size required to meet a pre-specified precision 92-96 Ex14.2 Ex14.4 perform calculations involving the difference between two population means 88-91 Ex14.5 Nulake Sigma I can Excellence External describe the applications of the Central Limit Theorem 73-77 Ex11.111.5 calculate the standard error of an estimate 73-77 181 analyse the estimates of population parameters (eg discuss the precision). 73-77 use sigma notation in a proof Use of appropriate technology is expected. 276 Achievement Standard Statistics and Modelling 3.3 Solve straightforward problems involving probability 4 Credits I can calculate probabilities involving mutually exclusive events calculate probabilities involving independent events Achievement: calculate probabilities involving complementary events External Nulake Sigma P108111,170 P108111,p170 Merit: Ex 1.3 P125-130 Ex1.3 calculate expected values or variance of random variables from a given discrete probability distribution. P146-159 Ex 1.1,2.22.10 calculate probabilities using theoretical and experimental probability P108-111 1 calculate probabilities using use tree diagrams P112-116 calculate probabilities using Venn diagrams P123-130 Ex 1.2 calculate probabilities using permutations and combinations P131-146 Ex 3.13.5 calculate probabilities involving combined events. use basic sigma notation Excellence Ex 1.3 Ex 1.4 P276 I can Nulake Sigma solve problems from those situations listed for achievement. P164-170 Ex 1.6 solve problems involving conditional probability P116-122 Ex 1.5,1.6 I can Nulake Sigma apply probability theory using combinations of techniques P164-170 apply probability theory using non-numeric reasoning P164-170 prove selected probability theories P164-170 use sigma notation if required in calculations or proofs. Use of appropriate technology is expected. Achievement Standard Statistics and Modelling 3.4 Solve equations 4 Credits Achievement: I can Sigma 172-178 182-187 Ex21.1, 21.2 solve a non-linear equation using the Newton-Raphson method with a given starting value (only derivatives of polynomials are required) 193-199 Ex23.4 solve a non-linear equation using the bisection method with a given starting interval 188-191 Ex23.3 optimise an objective function for a situation requiring techniques of linear programming, where the constraints and the objective function for the problem are given. 200-210 Ex22.4, 22.5 I can Nulake Sigma form and solve a system of linear equations. 182, 187 211-215 220-223 211-215 220-223 Ex21.2 -21.4 Ex22.3 22.5 Ex22.4 -22.5 find appropriate solutions to a non-linear equation using the Newton-Raphson method to improve the approximation to a stated precision or for a specified number of iterations. 216-219 Ex23.4 find appropriate solutions to a non-linear equation using the bisection method to improve the approximation to a stated precision or for a specified number of iterations. 216-217 Ex23.3 I can Nulake interpret the consistency or non–independence of systems of equations, 179-181 optimise an objective function for the linear programming problem. Merit: Nulake solve systems of three linear equations in three variables, where there is a unique solution form constraints and the objective function for a linear programming problem. consider sensible rounding of an answer to an linear programming problem use a graph, table or graphics calculator to find an approximate solution to a nonlinear equation represent geometrically non–independent systems of equations, Excellence External Sigma 346 347348 determine the effect of varying the parameters of a linear programming problem find multiple solutions, where possible, to a linear programming problem give the advantages and disadvantages of the Newton-Raphson method or the bisection method 378 derive the formulae for the Newton-Raphson method 375 give a geometric description of the Newton-Raphson method 375376 give a geometric description of the bisection method 372 Use of appropriate technology is expected. Achievement Standard Statistics and Modelling 3.5 Complete a statistical investigation involving bi-variate data 3 Credits Internal Achievement: I can Nulake Booklet Nulake Booklet Nulake Booklet Plan and carry out an investigation involving bi-variate continuous data, analyse the data and report the findings, including: a purpose statement identification of appropriate variables a description of the data collection method or data source Use regression to explore the relationship between pairs of variables. Write a report that is related to the context addresses the purpose of the investigation describes the relationship between at least one pair of variables I can carry out an in-depth analysis of bi-variate data within an investigation analyse bi-variate data by establishing a model using regression Merit: discussing the appropriateness of the model established interpret regression analysis and correlation coefficients (which may involve comparison of the relationship between more than one pair of variables) make predictions from regression equations (interpolation and/or extrapolation) improve regression models, eg consider the effect of outliers, fitting piecewise or nonlinear models understand the difference between correlation and causality. write a report that is consistent with the analysis. I can critically evaluate an investigation referring to limitations Excellence critically evaluate an investigation referring to possible improvements critically evaluate an investigation referring to justification of method critically evaluate an investigation referring to alternative approaches critically evaluate an investigation referring to assumptions made critically evaluate an investigation referring to potential sources of bias critically evaluate an investigation referring to relevance and usefulness of evidence critically evaluate an investigation referring to how widely the findings can be applied The use of appropriate technology for exploration of data and communication of findings is expected. This achievement standard does not assess sampling concepts or the use of confidence intervals (see AS90288, Mathematics 2.5, and AS90642, Statistics 3.2 Achievement Standard Statistics and Modelling 3.6 Achievement: Use probability distribution models to solve straightforward problems 4 Credits External I can Nulake Sigma solve straightforward problems using the binomial distribution P264-271 Ex 5.1-5.4 solve straightforward problems using the Poisson distribution P272-276 Ex 6.1-6.2 solve straightforward problems using the normal distribution P277-288 Ex7.1-7.3 calculate sample statistics to estimate for a binomial distribution Ex 5.5 calculate sample statistics to estimate for a Poisson distribution P276 calculate probabilities using formulae P266,273 Excellence Merit: use basic sigma notation Ex 6.3 Ex 5.1,6.1 Ex 16.6 I can Nulake use probability distribution models to solve problems involving sums, and other linear combinations, of normally distributed independent variables P304-306 Ex 7.6 solve inverse normal problems P289-293 Ex 7.4 I can Nulake Sigma use and justify probability distribution models to find a parameter given a probability P307-311 Ex7.4-7.5 Ex6.4,5.5 select and justifying the use of a model or approximation P307-311 Ex 8.4 use and justify probability distribution models to approximate a distribution P307-311 Ex 8.1-8.4 use a continuity correction where a discrete distribution is approximated by a continuous distribution P298-303 Ex 8.1,8.3 use sigma notation in calculations or proofs Sigma Achievement Standard Statistics and Modelling 3.7 Use a mathematical model involving curve fitting to solve a problem 3 Credits Internal Achievement: I can use y = ax Nulake n as a mathematical model to solve a problem given the model. use y = amx as a mathematical model to solve a problem given the model. use y = Aekt as a mathematical model to solve a problem given the model. I can use a power function as an appropriate mathematical model to solve a problem. use a exponential function as an appropriate mathematical model to solve a problem. Merit: p334335 graph the model select a model to test I can justify the choice of mathematical model to solve a problem. p312 p312 p330331 p312 Ch & Pg no. p334335 p330331 use a piecewise function as an appropriate mathematical model to solve a problem. determine the equation of the model Sigma p312 p312 p312 p337, 341342 p315347 p343347 p312 p312 p312 Ch & Pg no. p343347 Excellence relate the solution to the problem consider the nature of the underlying variables consider the possibility of more than one model p340342 relate the theory behind the model to the given context discussing limitations of the model derive the formula for a logarithmic transformation. p313 Chapter18 from Sigma gives good background information. Chapter 7 from Nulake is the relevant chapter for this work. Some worksheets and/or additional notes will be given.