§-Section

Unit 4

7 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does a sample proportion vary from sample to sample, and what distribution does it follow?

Understand random sampling and describe the distribution of the sample proportion, including its mean and standard deviation.

Why random sampling matters, how the sample proportion behaves as a random variable, and the mean and standard deviation of its approximately normal sampling distribution for TCE Mathematics Methods Unit 4.

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How do we reverse the derivatives of exponential and trigonometric functions to find their antiderivatives?

Find antiderivatives of exponential and trigonometric functions and apply them in definite integrals.

The standard antiderivatives of e^kx, sin and cos functions, why each carries a reciprocal factor, and how to apply them in definite integrals for TCE Mathematics Methods Unit 4.

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How do we estimate a definite integral when we cannot find an antiderivative?

Use the trapezoidal rule to approximate definite integrals and areas under curves.

How to approximate a definite integral with the trapezoidal rule, set up the calculation from a formula or a table of data, and judge whether the estimate is too high or too low for TCE Mathematics Methods Unit 4.

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How do we find the area enclosed between two curves rather than under a single curve?

Calculate the area enclosed between two curves using definite integration.

How to find the region enclosed between two graphs by integrating the difference of the upper and lower functions, including finding intersection limits and handling curves that cross, for TCE Mathematics Methods Unit 4.

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How do we describe continuous random variables and work with the normal distribution?

Use probability density functions and the normal distribution, including standardisation to z-scores, to find probabilities.

A continuous random variable is described by a probability density function whose area gives probability; the normal distribution is the key model, and standardising to z-scores lets you compute its probabilities.

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How do we estimate a population proportion from a sample and express the uncertainty?

Use the distribution of sample proportions to construct and interpret confidence intervals for a population proportion.

Sample proportions vary from sample to sample with an approximately normal distribution; a confidence interval uses this to give a plausible range for the unknown population proportion at a stated level of confidence.

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What is integration, how is it the reverse of differentiation, and how do we find areas?

Find antiderivatives and definite integrals and use them to calculate areas and solve rate problems.

Integration is antidifferentiation; the fundamental theorem of calculus links the definite integral to the antiderivative, letting you compute areas under curves and recover quantities from their rates of change.

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