Cambridge A-Level Mathematics
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Pure Mathematics 1
Quadratics
1.1 - Completing the squareTwo expressions can describe the same quadratic and reveal different things about it. In this first part of 1.1, you will rewrite a quadratic as a square plus a constant, then use that form to find its turning point and sketch its graph. The method also explains why some quadratics can never be n...
1.1 - Solving quadratic equationsA quadratic equation may have two different real solutions, a repeated solution, or no real solution. You will solve it by factorising, completing the square and using the quadratic formula, then choose a method that fits the equation. Exact answers will let you check the algebra without rounding.
1.1 - Counting real rootsYou can often decide how many real solutions a quadratic has without finding either solution. You will use the discriminant, apply it to equations containing a parameter and connect the result to where graphs cross or touch. The distinction between two roots and a repeated root will matter throug...
1.1 - Solving quadratic inequalitiesAn equation finds the inputs where a quadratic is zero; an inequality asks where it is positive, negative or equal to a boundary. You will use roots to divide the number line into intervals, handle strict and inclusive endpoints and recognise when every real input, or none, satisfies an inequality.
1.1, 1.3 - Linear and quadratic intersectionsA point common to two curves must satisfy both equations at once. You will use substitution to solve a linear and a quadratic equation, including equations with circles or products of coordinates. You will then connect the resulting quadratic to the graph relationship in 1.3: crossing, touching o...
1.1 - Equations in quadratic formAn equation can hide a quadratic inside powers, square roots or reciprocals. You will identify a useful replacement variable, solve the resulting quadratic and return carefully to the original input. Conditions on the replacement variable will explain why some algebraic candidates must be rejected.
Functions
1.2 - Functions, domains and rangesA function is more than a formula: its permitted inputs decide which outputs are possible and whether the rule can be reversed. In this part of 1.2, you will use mappings and familiar graphs to find domains and ranges, distinguish one–one functions from many–one functions, and choose useful domai...
1.2 - Combining functionsApplying one function after another creates a new function, but the order and permitted inputs both matter. In this part of 1.2, you will form composite rules, decide when a composition exists, and solve equations while keeping the original domain restrictions.
1.2 - Reversing a functionAn inverse function recovers an input from its output, so the original function must be one–one. In this part of 1.2, you will rearrange rules to find inverses, use domain restrictions to choose the correct branch, and reflect graphs in the line y=x.
1.2 - Moving and stretching graphsA graph can be moved, reflected or stretched without calculating a new table of values. In this part of 1.2, you will track its points to understand changes inside and outside a function, describe transformations precisely, and combine them in the correct order for algebraic and given-feature gra...
Coordinate geometry
1.3 - Equations of straight linesHow can two coordinates determine a line that continues for ever? In this part of 1.3, you will connect a line's direction to its equation, then use distances, midpoints and intersections to solve geometry problems. Horizontal and vertical lines will help explain the limits of the familiar gradie...
1.3 - Equations of circlesA circle is made of points that are all the same distance from one fixed point. In this part of 1.3, that simple description will become an equation, and completing the square will uncover a hidden centre and radius. You will also use diameters, right angles and symmetry to connect circle geometr...
1.3 - Lines meeting circlesHow can algebra distinguish a line that cuts a circle twice from one that just touches it? In this part of 1.3, you will find intersection coordinates, construct a tangent using its radius, and use a repeated root to determine an unknown tangent. The same equations explain all three possibilities...
Circular measure
1.4 - Measuring angles in radiansAn angle can be measured by comparing a curved length with a straight one. In this part of 1.4, you will define radians, convert between radians and degrees, and use radians to calculate arc lengths. The new unit makes a circle's geometry especially simple: arc length becomes radius multiplied by...
1.4 - Areas and perimeters of sectorsA curved region's area and its perimeter come from different pieces of geometry. In this part of 1.4, you will derive the sector-area formula, separate a segment from its triangle, and trace boundaries involving arcs and straight edges. The same chord will lead to two very different areas: a smal...
Trigonometry
1.5 - Exact values and reference anglesWhy can two different angles have the same sine, yet a calculator return just one of them? This part of 1.5 develops exact trigonometric values from two simple triangles, extends them around the unit circle, and explains what an inverse-trigonometric answer means. You will work in both degrees an...
1.5 - Sine, cosine and tangent graphsA moving point on a circle produces a repeating wave when you record one coordinate against its angle. In this part of 1.5, you will sketch sine, cosine and tangent, then change their sizes, periods and positions. You will also work backwards from a graph's features to an equation and use interse...
1.1, 1.5 - Trigonometric identities and equationsA calculator gives one inverse-trigonometric angle, but an equation may have several roots in its interval. This lesson combines the quadratic methods of 1.1 with the identities and repeating graphs of 1.5. You will simplify and prove identities, find every permitted angle, and turn mixed trigono...
Sequences and series
1.6 - Expanding integer powersMultiplying a bracket by itself many times produces a pattern you can use without carrying out every multiplication. In this part of 1.6, you will build the binomial coefficients, expand positive integer powers and find selected coefficients, including in a product of expressions.
1.6 - Arithmetic progressionsAdding the same amount each time creates a sequence with a simple structure. In this part of 1.6, you will recognise an arithmetic progression, find its terms and totals, and use equations to compare two progressions. The distinction between one term and a sum will guide every calculation.
1.6 - Geometric progressionsMultiplying by the same number repeatedly creates a geometric progression. In this part of 1.6, you will recognise this structure, calculate terms and finite sums, and connect a geometric progression with an arithmetic one. A negative multiplier is allowed, and a common value need not occupy the...
1.6 - Sums to infinityAn endless sum can have a finite value, provided its partial totals settle towards one number. In this final part of 1.6, you will explain when a geometric series converges, calculate its sum to infinity and use the convergence condition to accept or reject possible ratios. Alternating signs can...
Differentiation
1.7 - From chords to a tangentA curve does not have one gradient for its whole length. You will find its gradient at a particular point by bringing a second point closer, then connect this idea to first and second derivatives. This gives a precise meaning to the slope you see in a sketch.
1.7 - Differentiating powersChord gradients explain what a derivative means; rules make it quick to find one. You will differentiate rational powers, rewrite roots and fractions before differentiating, and find second derivatives. Keeping the original domain in view will prevent an algebraic answer being used where the func...
1.7 - Differentiating a function of a functionThe expression inside a power may itself change with x. You will use the chain rule to follow both stages of that change, including roots and reciprocal powers with nonlinear expressions inside. Expanding simple cases will help you check that no factor has been missed.
1.7 - Tangents and normalsA derivative supplies the gradient needed for a straight-line equation. You will find tangents and perpendicular normals, handle horizontal and vertical cases, and work backwards from a line condition to an unknown contact point. You will then use ordinary line intersections to finish geometric p...
1.7 - Rates of changeA derivative measures how one quantity changes as another changes. You will interpret its sign and units, connect two changing quantities through the chain rule, and find intervals where a function increases or decreases. These ideas turn gradients into useful information about a model.
1.7 - Stationary points and optimisationA zero gradient identifies a stationary point, but it does not tell you whether the point is a maximum or a minimum. You will locate stationary points, classify them with second derivatives or gradient signs, and use that information in sketches and optimisation. Domains and boundary values will...
Integration
1.8 - Integration as reverse differentiationDifferentiation turns a curve into its gradient function. Integration reverses that process, but a gradient cannot tell you how far up the curve sits. You will recover families of functions, integrate rational powers and linear brackets, and check each result by differentiating it.
1.8 - Finding a curve from its gradientA gradient function determines a family of curves; one point can select a particular member. You will find a curve's equation by integrating and using given information, recover a function from its second derivative, and check that every condition has been satisfied.
1.8 - Definite integralsAn indefinite integral is a family of functions; a definite integral has limits and gives a value. You will evaluate definite integrals exactly, handle simple finite and infinite endpoints with limits, and use a given integral to find an unknown boundary.
1.8 - Areas between curvesAn integral adds signed contributions, whereas a geometrical area is non-negative. You will split a region when its boundary crosses an axis, find intersections before subtracting curves, and choose vertical or horizontal strips to calculate an enclosed area.
1.8 - Volumes of revolutionRotating a flat region about an axis creates a solid. You will build its volume from circular cross-sections, choose the correct radius and limits for either coordinate axis, and subtract an inner volume when the solid has a hole.
Advanced pure mathematics
Algebra
2.1, 3.1 - Modulus graphs and equationsTwo numbers can have the same distance from zero without being equal. In this shared part of 2.1 and 3.1, you will turn that observation into graphs, equations and inequalities involving modulus. Splitting a straight line at its zero will make each problem ordinary linear algebra.
2.1, 3.1 - Dividing polynomialsDividing a polynomial produces a quotient and, sometimes, something left over. In this shared part of 2.1 and 3.1, you will organise division by linear and quadratic expressions, including missing powers and non-unit leading coefficients. Multiplication will provide a complete check of every result.
2.1, 3.1 - Factor and remainder theoremsOne substitution can reveal a remainder without carrying out a whole division. In this shared part of 2.1 and 3.1, you will explain why that works, use factor and remainder conditions to find unknown coefficients, and reduce polynomial equations to familiar lower-degree equations.
Logarithms and exponentials
2.2, 3.2 - Logarithms and their lawsA logarithm answers a power question: which exponent produces a given number? In this shared part of 2.2 and 3.2, you will translate between logarithms and indices, combine logarithms using their laws, and solve equations while keeping every logarithm's input positive.
2.2, 3.2 - The functions exp and lnThe natural exponential and natural logarithm undo one another. In this shared part of 2.2 and 3.2, you will connect their graphs, solve equations with unknown exponents and handle exponential inequalities, including decreasing powers. The sign of a logarithm will explain when an inequality rever...
2.2, 3.2 - Straightening curved relationshipsA curve can become a straight line when its axes use logarithms. In this shared part of 2.2 and 3.2, you will transform power and exponential relationships, connect gradients and intercepts to their constants, and recover the original model from a transformed graph. The axis labels will decide wh...
Advanced trigonometry
2.3, 3.3 - Reciprocal trigonometric functionsTaking the reciprocal of a trigonometric function changes its graph and its permitted inputs. In the shared trigonometry of 2.3 and 3.3, you will use secant, cosecant and cotangent, explain their asymptotes, and derive identities that turn unfamiliar equations into familiar quadratics.
2.3, 3.3 - Compound-angle formulaeKnowing sine and cosine of two angles lets you find sine and cosine of their sum, but you cannot simply add the ratios. This shared lesson for 2.3 and 3.3 develops the compound-angle formulae, uses them for exact values and proofs, and solves equations whose angles have been shifted.
2.3, 3.3 - Double-angle identitiesA double-angle identity connects a function of twice an angle to functions of the original angle. In this shared lesson for 2.3 and 3.3, you will derive the identities, choose the form that suits the information given, and solve equations without losing roots or accepting undefined expressions.
2.3, 3.3 - Combining sine and cosineA sum of sine and cosine with the same angle is another shifted sine or cosine. In the shared trigonometry of 2.3 and 3.3, you will find its amplitude and angle shift, use them to determine attainable extreme values, and solve equations in a specified interval.
Advanced differentiation
2.4, 3.4 - Differentiating exponentials and logarithmsThe height of an exponential curve also tells you its gradient. You will use that special property of e^x, differentiate natural logarithms and apply the chain rule to find rates of change and tangent equations.
2.4, 3.4 - Differentiating trigonometric functionsA sine curve is steepest where it crosses its centre line and flat at its peaks. You will connect that picture to the derivatives of sine, cosine and tangent, then differentiate composite functions and locate stationary points within a stated interval. All calculus angles in this lesson are in ra...
2.4, 3.4 - Product and quotient rulesTwo changing factors contribute to the change in their product. You will learn how to account for both contributions, differentiate fractions without losing the subtraction order, and combine these rules with the chain rule for logarithmic, exponential and trigonometric functions.
2.4, 3.4 - Differentiating parametric curvesA curve can be described by giving both coordinates in terms of a third variable. You will follow such a curve, turn its two parameter derivatives into a gradient, and find tangents and normals without first finding an equation of the form y=f(x).
2.4, 3.4 - Implicit differentiationAn equation can describe a curve without making y its subject. You will differentiate both sides while treating y as dependent on x, collect the resulting derivative terms, and find tangent and normal equations, including horizontal and vertical cases.
Advanced integration
2.5, 3.5 - Exponential and trigonometric integralsIntegration reverses differentiation, but a function inside another function leaves a constant factor to undo. In 2.5 and 3.5, you will use that idea to integrate exponentials, reciprocals of linear expressions and three standard trigonometric forms. Checking by differentiation will keep the fact...
2.5, 3.5 - Using identities to integrateSome trigonometric integrands do not look like any standard derivative until you rewrite them. In 2.5 and 3.5, double-angle and reciprocal identities turn squared functions into forms you can integrate. You will then choose suitable identities for exact definite integrals and distinguish a signed...
Numerical equations
2.6, 3.6 - Locating a rootAn equation can have a perfectly definite root without having a convenient exact expression for it. In 2.6 and 3.6, graphs will suggest where to look, and calculated signs will place a root inside a chosen interval. You will also show why a rounded answer is accurate and recognise when a sign cha...
2.6, 3.6 - Iterating towards a rootA root bracket tells you where a solution lies; iteration gives a practical way to calculate increasingly accurate approximations. In 2.6 and 3.6, you will connect a recurrence to its original equation, use it with retained calculator precision and recognise failed attempts. A final sign-change c...
Pure Mathematics 3 extensions
Algebraic extensions
3.1 - Partial fractionsAdding fractions creates a single rational expression; partial fractions reverse that process. In this part of 3.1, you will split fractions with distinct linear factors, a repeated factor or an irreducible quadratic. Choosing the right numerator forms will turn each decomposition into ordinary a...
3.1 - Binomial series with rational powersA positive integer power gives a finite binomial expansion; a negative or fractional power usually continues indefinitely. In this part of 3.1, you will calculate the first few terms, adapt the standard form and state its validity interval. You will also multiply short expansions and use them to...
Calculus extensions
3.4, 3.5 - Inverse tangent in calculusInverse tangent connects a familiar angle function to a new kind of rational integral. In this lesson, 3.4 supplies its derivative and the chain rule extends it to composite functions; 3.5 reverses that derivative to integrate positive quadratic denominators. Principal radian values will then giv...
3.5 - Recognising a logarithmic integralSome fractions hide a familiar derivative: the numerator is a multiple of the denominator's derivative. In this part of 3.5, you will recognise that structure, adjust constants or rearrange a numerator, and integrate trigonometric examples such as tangent. Absolute values and interval checks will...
3.5 - Integrating partial fractionsA rational function can hide several familiar integrals inside one fraction. In this part of 3.5, you will decompose it first, then recognise logarithmic, power and inverse-tangent terms. You will also evaluate exact definite integrals and check that their intervals do not cross an undefined input.
3.5 - Integration by partsThe product of two easy functions may be awkward to integrate. Integration by parts reverses the product rule and replaces one integral with another, chosen to be simpler. In this part of 3.5, you will make that choice, repeat the method when necessary and uncover products hidden inside logarithm...
3.5 - Integration by substitutionA change of variable can turn an unfamiliar integral into a familiar one. In this part of 3.5, you will use a supplied substitution to change every part of the integral, including its differential and, when appropriate, its limits. Algebraic and trigonometric examples will show why keeping the ol...
Vectors in space
3.7 - Vectors and coordinatesA journey has a direction as well as a length. In this part of 3.7, you will describe directed movements in two and three dimensions, connect points using position vectors, and calculate lengths and unit vectors. The same language will let us describe entire lines in space.
3.7 - Vector equations of linesTwo lines in a flat diagram usually meet unless they are parallel. In space, they can pass each other without meeting. In this part of 3.7, you will describe a line using a point and a direction, test whether points lie on it, and distinguish parallel, intersecting and skew lines.
3.7 - Angles and perpendiculars with vectorsCoordinates can reveal an angle that is hard to see in a drawing of a solid. In this final part of 3.7, you will use scalar products to find angles and perpendicular directions, then locate the nearest point on a line. Choosing the direction of each vector will matter as much as the calculation.
Differential equations
3.8 - Solving separable differential equationsKnowing a rate of change can tell us how a quantity develops, even when we do not yet know its formula. In this part of 3.8, you will translate rate statements into differential equations, separate variables and integrate, then use an initial value to select a particular solution. Checking the or...
3.8 - Interpreting differential-equation modelsTwo models can agree with the same starting value and initial rate, yet make different predictions later. In this final part of 3.8, you will build simple rate models, solve and interpret them, and judge where their predictions remain meaningful. The context will supply the assumptions; no specia...
Complex numbers
3.9 - Calculating with complex numbersThe equation x²+1=0 has no real solution, but it becomes solvable when the number system is enlarged. You will calculate with the new number i, separate real and imaginary parts, and use conjugate pairs to recover all the roots of a polynomial with real coefficients.
3.9 - Argand diagrams and polar formA complex number can be located by two coordinates or by a distance and a direction. You will use an Argand diagram to connect these descriptions, interpret addition and subtraction as displacements, and convert between Cartesian, polar and exponential notation.
3.9 - Rotations and square rootsMultiplying by i turns a point through a right angle: can a calculation really perform a rotation? You will multiply and divide using moduli and arguments, interpret complex operations geometrically, and find both square roots of a complex number with exact Cartesian working.
3.9 - Loci in an Argand diagramAn equation in a complex number can describe an entire set of points. You will read modulus conditions as distances, equal moduli as equal distances, and an argument condition as a direction from a fixed point. Combining those ideas will let you describe regions and keep their boundaries precise.
Pure Mathematics 2 only
Mechanics
Forces and equilibrium
4.1, 4.4 - Modelling forcesA box can have several forces acting on it even while it stays still. You will turn physical situations into particle models, distinguish mass from weight, and identify the two bodies in a Newton's third-law pair. These ideas from 4.1 and the mass–weight relationship in 4.4 are the starting point...
4.1 - Components and resultantsTwo forces can point in quite different directions yet combine into one force. You will resolve forces into perpendicular components, calculate a resultant's magnitude and direction, and find unknown forces that keep a particle in equilibrium. The diagram helps you choose the equations; calculati...
4.1 - Friction and limiting equilibriumA gentle pull on a rough surface may produce no motion at all: friction can adjust to the force needed for balance. You will determine its direction and magnitude, check whether rest is possible, and use limiting equilibrium when a particle is about to slip. Resolving perpendicular to the surface...
Motion in a straight line
4.2 - Displacement and velocity graphsA particle can travel a long way and finish back where it started. You will distinguish total travel from signed change of position, read velocity and acceleration from graph gradients, and find displacement and distance from a velocity–time graph. Throughout this lesson, motion is along one stra...
4.2 - Constant accelerationAt the highest point of a vertical throw, velocity is zero but gravitational acceleration is still acting. You will use the equations for constant acceleration, keep direction signs consistent, and connect stages or two particles with the correct times. These methods calculate motion without firs...
4.2 - Motion with calculusWhen acceleration changes with time, a single constant-acceleration equation no longer describes the whole motion. You will differentiate position to obtain velocity and acceleration, integrate rates using initial conditions, and split a journey at genuine direction changes to find total distance...
Newton's laws
4.4 - Forces causing accelerationThe direction a particle moves need not be the direction it accelerates. You will use force diagrams to calculate acceleration, resolve motion on rough planes, and connect vertical force equations to constant-acceleration motion. The resultant force, rather than any one force, determines how velo...
4.4 - Connected particlesA string or tow-bar links the motions of different bodies, but each body still has its own forces. You will use the modelling assumptions behind strings and pulleys, choose between whole-system and individual equations, and distinguish a connector pulling in tension from a rigid connector pushing...
Energy and power
4.5 - Work and energyA force can act on a moving body yet do no work, or do negative work by opposing its displacement. You will calculate work using the force component along a displacement, find kinetic and gravitational potential energies, and build energy balances that include driving forces and resistance. Choos...
4.5 - Conservation of energySpeed can sometimes be found without calculating acceleration or time. You will choose an energy system, conserve mechanical energy when only gravity transfers energy between motion and height, and include the work against resistance when energy is lost. The same balance handles a straight slope...
4.5 - Power and motionAn engine's power tells you how quickly it transfers energy, rather than the size of its driving force alone. You will calculate average power from work and time, use speed to find an instantaneous driving force, and combine that force with resistance and gravity to calculate a vehicle's accelera...
Probability and Statistics 1
Representing data
5.1 - Choosing a data displayA list preserves every observation, but a well-chosen display reveals patterns much faster. In this part of 5.1, you will choose a display for a purpose, organise values in stem-and-leaf diagrams, and construct and interpret histograms. The central question is what each picture preserves and what...
5.1 - Cumulative frequency and quartilesA grouped table can tell you how many observations lie below a boundary even when the individual values are unknown. In this part of 5.1, you will build cumulative frequency graphs, find medians and quartiles, and compare datasets using box-and-whisker plots. You will distinguish exact summaries...
5.1 - Mean and standard deviationTwo datasets can have the same centre but very different spreads, and one unusual observation can change some summaries far more than others. In this part of 5.1, you will choose and calculate measures of location and spread, find means and standard deviations from frequencies, and work with code...
Counting arrangements
5.2 - Permutations and combinationsChoosing three people and giving those people three different jobs produce different counts. In this first part of 5.2, you will decide when order matters, count ordered and unordered selections, and combine simple restrictions without counting the same outcome twice.
5.2 - Arrangements with restrictionsAn arrangement count changes when some objects look identical or some positions are restricted. In this part of 5.2, you will remove duplicate arrangements, keep specified objects together or apart, and count seating in several rows by treating each seat as a distinct position.
Probability
5.3 - Combining eventsTwo fair dice have eleven possible totals, but those totals are not equally likely. In this first part of 5.3, you will count equally likely elementary outcomes, combine alternatives without double counting, and distinguish independent events from mutually exclusive events.
5.3 - Conditional probabilityA machine may produce most of a factory's items without producing most of its faulty items. In this part of 5.3, you will update a probability when information restricts the sample space, use tree diagrams for successive choices, and reverse a condition by keeping the correct denominator.
Discrete distributions
5.4 - Discrete random variablesA random experiment can have many outcomes but only a few possible numerical scores. In this first part of 5.4, you will collect those scores into a probability distribution, calculate their long-run mean and variance, and recover unknown probabilities from the distribution's summaries.
5.4 - The binomial distributionWhen the same chance experiment is repeated a fixed number of times, how can you find the chance of a particular number of successes? Learn when a binomial model fits, calculate exact and cumulative probabilities, and use its mean and variance to recover unknown parameters.
5.4 - Waiting for the first successRepeating trials until the first success gives a waiting time rather than a fixed-trial success count. Learn to recognise a geometric distribution, count the failures in point and tail probabilities, and interpret its expected waiting time.
Normal distribution
5.5 - Normal probabilitiesA normal curve turns a measurement range into an area. Learn what its parameters mean, standardise measurements to use the supplied normal tables, and combine left-tail areas to find between and outside probabilities.
5.5 - Finding normal cut-offs and parametersSometimes the area under a normal curve is known and the measurement is missing. Work backwards from a probability to a standardised value, find cut-offs, and use one or two probability statements to determine an unknown mean or standard deviation.
5.5 - Approximating a binomial distributionA binomial probability may involve many terms. Learn when a normal curve is a suitable approximation, use half-integer boundaries to match a discrete count to a continuous area, and check that the final approximation describes the intended event.
Probability and Statistics 2
Poisson models
6.1 - The Poisson distributionA random count can have a steady average while varying considerably from one interval to the next. Learn when a Poisson model is appropriate, calculate probabilities over different intervals, and use its single parameter to describe both its centre and spread.
6.1 - Approximating with Poisson and normal modelsA rare-event binomial count can often be approximated by a Poisson distribution, while a large-mean Poisson count can often be approximated by a normal curve. Learn to justify the choice, match its parameters and handle the event boundaries correctly.
Combining random variables
6.2 - Means and variances of combinationsMultiplying one random measurement by four and adding four independent measurements can give the same expected value but different spreads. Learn how shifts, scales, sums and differences affect expectation and variance, and distinguish repeated observations from repeated use of the same observation.
6.2 - Normal and Poisson combinationsSome distributions retain their form when random variables are combined. Learn to find probabilities for transformed normal measurements, independent normal sums and differences, and independent Poisson totals, while keeping the conditions for each result clear.
Continuous distributions
6.3 - Probability density functionsA continuous measurement can have zero probability at every single exact value and still have total probability one. In this first part of 6.3, you will explain that apparent puzzle using area, find probabilities from a density function, and handle distributions whose possible values extend indef...
6.3 - Mean, variance and percentilesA density describes a whole distribution, so it can tell you much more than the probability of one interval. In this part of 6.3, you will calculate its mean and variance, locate medians and other percentiles by area, and distinguish the distribution's average from its regions of greatest density.
Sampling and estimation
6.4 - Random samples and unbiased estimatesYou rarely measure every member of a population, so a sample has to support conclusions about a larger group. In this first part of 6.4, you will distinguish parameters from statistics, choose samples with random numbers, recognise sources of bias and calculate unbiased estimates of a population...
6.4 - The distribution of a sample meanIndividual measurements vary, and so do the means of repeated samples. In this part of 6.4, you will find the centre and spread of a sample mean, use its exact normal distribution for a normal population, and apply the Central Limit Theorem when a large sample makes normal approximation appropriate.
6.4 - Confidence intervals for a meanA sample mean estimates one unknown number, but different random samples give different estimates. In this part of 6.4, you will turn that sampling variation into a confidence interval, interpret its confidence level, and calculate how confidence, population spread and sample size affect precision.
6.4 - Confidence intervals for a proportionA survey's sample proportion is an estimate, not the exact population proportion. In this final part of 6.4, you will calculate its estimated standard error, build a large-sample confidence interval and explain how confidence and sample size affect the precision of a reported percentage.
Hypothesis testing
6.5 - Hypotheses and critical regionsAn unusual result might reflect a real change, or it might be ordinary random variation. You will turn a claim about a population into hypotheses, choose a rejection rule before observing the data, and explain what the resulting decision means. The rule controls how readily chance alone can lead...
6.5 - Binomial and Poisson testsA very small count may suggest a reduced probability; a very large count may suggest an increased event rate. You will calculate the relevant null tail for binomial and Poisson data, find discrete critical boundaries, and use a normal approximation when its conditions support it. The observation...
6.5 - Testing a population meanA sample average can differ from a claimed population mean even when that claim is correct. You will judge the size of that difference using the distribution of a sample mean, carry out one-tailed and two-tailed normal tests, and connect probability comparisons with critical boundaries. The sprea...
6.5 - Type I and Type II errorsA test can reject a true claim or miss a real change. You will distinguish these two mistakes, calculate their probabilities from a fixed decision rule, and see why demanding stronger evidence can make a genuine change harder to detect. The distribution used in an error calculation depends on whi...