75 items found for ""
- Acceleration, Projectiles & Kinematics
Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction). velocity = distance / time The units of velocity are m/s Acceleration is the rate of change of velocity, and so is the mathematical derivative of this. acceleration = change in velocity / change in time The units of acceleration are m/s² Motion Graphs We can plot motion on two main types of graph - it is important to know the properties of each. Displacement-Time Graphs The Gradient is the velocity - draw a tangent to find the instantaneous velocity Horizontal line represents zero velocity Velocity-Time Graphs The Gradient is the acceleration The Area beneath the graph is the displacement Constant Acceleration When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables: s is for displacement u is for initial velocity v is for final velocity a is for acceleration t is for time Vertical Motion due to Gravity The gravitational force of the earth causes all objects to accelerate towards the ground, its surface. Ignoring air resistance, the acceleration is constant, and given a g: g = 9.8 m/s² This is independent of the mass, shape or velocity of the object. It is vital to set a positive direction of motion for each question In the example above, we set up as the positive direction, so our values for a and v were negative because the ball is going down. Constant Acceleration with Vectors Additionally, we can express motion using vectors: r = r₀ + v t r is the position vector of the moving object r₀ is the initial position vector v is the velocity vector Four of the five SUVAT equations have vector equivalents: v = u + at v = u + a t s = ut + ½at² s = u t + ½ a t² + r₀ s = vt - ½at² s = v t - ½ a t² + r₀ s = ½(u+v)t s = ½( u + v )t + r₀ v² = u² + 2as has no vector equivalent Projectile Motion When we model a projectile, we ignore air resistance. This means that: Horizontal motion of a projectile has constant velocity: a = 0 Vertical motion of a projectile motion is modelled as gravitational free fall: a = g For horizontal projection, like in the diagram above: Since horizontal velocity is constant, we can use the equation x = vt (were x is horizontal displacement) For vertical velocity, we need to use SUVAT equations, due to the constant acceleration of g = 9.8 m/s² Horizontal and Vertical Components When given the velocity as a vector or at an angle, you must use trigonometry to find the vertical and horizontal components. Then, treat them separately as above (the horizontal component still has constant velocity) For an object projected at velocity U at an angle of θ to the x-axis: the horizontal component of the velocity is given as U cos(θ) the vertical component of the velocity is given as U sin(θ) The vertical component is decelerating at -9.8 m/s² as it rises, and accelerates at 9.8 m/s² as it falls The projectile reaches its maximum height when the vertical component of the velocity is zero. Applying SUVAT to the vertical component of the projectile gives us a few standard and useful equations: Projectile motion can also be plotted and calculated with in vectors. Variable Acceleration As you can see from the velocity-time graph, varying acceleration produces curves: This is because the gradient of a velocity-time graph is the acceleration, and so if the acceleration changes with time, so must the gradient. Kinematics Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity. Therefore: This means we can differentiate and integrate equations for motion: Integrating and Differentiating Vectors Variable acceleration can also be expressed in vector form. Therefore, you need to be able to differentiate and integrate vector equations. Often, dot notation is used to quickly represent differentiation with respect to time: To integrate vectors: In both differentiation and integration of vectors, you must do one term at a time See Notes Sheets on differentiation and integration in Pure Maths if you need a recap.
- Modelling in Mechanics
The real world of mechanics is incredibly complicated, with hundreds of different factors affecting motion and stability. This would be near impossible to calculate at this stage, so to simplify it we model objects and scenarios in a number of different ways. Modelling Assumptions Particle Has negligible dimensions Mass acts about a single point Rotational forces and air resistance can be ignored Rod The diameter is negligible (so no thickness), only length counts Mass acts along a line It is rigid, so does not bend or buckle Lamina An object with only area, thickness is negligible (like a sheet of paper) Mass acts across the flat surface Uniform Body The Mass is distributed evenly throughout the object The mass is modelled to act through one point, the centre of mass, in the geometrical middle of the body Light Object The mass of the object is negligible Often used for strings or pulleys Allows us to model the tension on each end of a string as equal Inextensible String A string that does not stretch when a force is applied Allows us to model the acceleration of two connected particles as the same Smooth Surface There is no friction between the surface and any object on it Rough Surface There is friction between the surface and any object on it Wire A rigid length of metal with negligible thickness (only modelled with length) Smooth and Light Pulley The pulley has no mass No friction in the pulley, so tension is the same either side of it Bead A particle with a hole in it for string/wire to be threaded through It can move freely along the wire or string Peg An object from which a body can be hung or rested Modelled as dimensionless and fixed Can be either rough or smooth Air Resistance Resistance forces due to motion through air Almost always modelled as negligible Gravity The force of attraction between all objects Acceleration due to gravity is given by g, which is 9.8 m/s² unless otherwise stated Assume objects are always attracted to the ground (earth) Gravity acts uniformly and downwards Quantities & Units Quantities can be either scalar or vector. Scalar quantities have only magnitude Vector quantities have both magnitude and direction SI Base Units Quantity Unit Symbol Mass kilogram kg Distance metre m Time seconds s Distance is scalar - the vector version of it is displacement Derived Units Quantity Unit Symbol Velocity metres per second m/s Acceleration metres per second² m/s² Force Newtons N Velocity is the vector form of speed, as velocity has direction as well as magnitude Weight is not the same as mass. Weight is a force, measured in Newtons, N. It is given by: W = mg Weight = mass x gravity Representing Forces Weight acts down from an object's centre of gravity. Other forces can act in any direction, and so to avoid confusion it is best to draw a force diagram: The length of arrow should represent the magnitude of the force. Single headed arrows represent forces. If a system is in equilibrium - meaning there is no net resultant force and it is motionless or at constant velocity - then all the force arrows will make a closed shape. The example above would give a triangle.
- Forces, Friction & Motion
As seen in the Notes Sheet on Modelling in Mechanics, forces on an object can be expressed with a force diagram: Newton's 1st Law: On this diagram, you can see that T, the tension force pulling the object to the right, is equal to F, the friction between the the object and the surface - friction is the force that opposes motion. This means that there is no resultant force to the left nor right, so the object remains motionless (it is in equilibrium). This is Newton's first law: An object will stay at a constant velocity, or motionless, unless it experiences a resultant force. Newton's 2nd Law: Here, the surface is smooth (no friction). This means the object experiences a resultant force, T, to the right. According to Newton's second law, the resultant force on an object is directly proportional to the rate of change of momentum of the object, and acts in the same direction. This is commonly expressed as: F = ma Resultant Force = Mass x Acceleration (in direction of resultant force) Newton's 3rd Law: In this example, the object sits at rest on the surface with no horizontal forces acting on it. However, its weight, W, (mass x g) is a force acting down. If this were the only force, you would expect the object to accelerate down through the surface but it does not. This means there is an equal but opposite force acting upwards: the normal reaction, R. For every action, there is an equal but opposite reaction (equal in magnitude and type of force). Forces in Two Dimensions Often, forces will not be exactly horizontal or vertical. Instead, they may be given as vectors or at angles. Sometimes you will need to find the resultant from two forces at right angles to one another, but more often you will be given the resultant and need to calculate the components. Use Pythagoras to do this. When given the resultant at an angle, use trigonometry to find the components. Moments Moments are the turning forces acting on an object, around a pivot. M = Fx moment = force × perpendicular distance from pivot The units of moments are Nm If there is a resultant moment in either direction (clockwise or anti-clockwise), then the object will rotate in that direction about its pivot. When an object is in equilibrium, there is neither a net force nor a net moment in any direction. The Principle of moments can be used for this: the sum of all clockwise moments must equal the sum of all anticlockwise moments for an object to be in equilibrium. When we model a rod or object as uniform, it means we assume all the weight is evenly distributed throughout, and only acts about the centre. Tilting The point of tilting is the point at which a rigid body is about to pivot. When a rigid body is at the titling point, the reaction at any other support is zero. Friction Friction is the force that opposes motion. No materials are perfectly smooth, but have a degree of roughness: this is defined by the coefficient of friction, μ, which is a fixed property of a surface in certain conditions (conditions matter, because if a surface is wet, its coefficient of friction will be lower). An object will only move on a surface if the force applied to it is greater than the maximum friction, F(max): this value is knows as the limiting value. The limiting value, F(max), depends on both the normal reaction force, R, and the coefficient of friction, μ: F(max) = μR Limiting Value = maximum friction = μR Static Rigid Bodies A rigid body is something like a rod or a ladder, leaning against a wall. If it is static, it is in equilibrium. For a rigid body to be in equilibrium: It must be stationary There must be no resultant force in any direction There must be no resultant moment A common example is a ladder leaning against a wall, where there is friction on the ground: Connected Particles & Pulleys When the particles are moving along the same straight line, they can be modelled as one particle with a combined mass. When the particles are moving separately, for example in a pulley system, they can be treated as separate particles. In both instances, when the string is light and inextensible, the acceleration in each particle will be the same. A common question will consist of two connected particles, one on an inclined plane, the other hanging off a pulley:
To start off with, there is some terminology you need to be familiar with. A population is the whole set of items that are of interest A sampling unit is an individual unit of a population A sampling frame is a named or numbered list of the sampling units in the population. A quantitative variable is one associated with numerical observations A qualitative variable is one that is non-numerical A continuous variable can take any value, e.g. decimals A discrete variable can only take fixed values, e.g. integers or colours A census measures every member of a population A sample is a selection of observations from a subset of the population There are advantages and disadvantages to all forms of statistical investigations: A census is entirely accurate (because it measures every sampling unit), but it is time consuming, cannot work with destructive testing (when the sample is destroyed when testing), and produces a vast amount of data to be processed. A Sample is less time consuming because less has to be tested and less data is produced, however it may not be as accurate, and the sample may not reflect the population well. Sampling Broadly speaking, there are two types of sampling - random, and non-random. These each have their own sub types, too. Random Sampling In random sampling, each member of a population has an equal chance of being chosen. This means the sample should be both representative and unbiased. Simple Random Sampling For a simple random sample, a sampling frame is created where each member is given a number. Then, a random number generator or a lottery is used to create the sample. Advantages are that it is free from bias, easy and cheap to use on small populations and samples, and the probability of being selected is known. Disadvantages are that a sampling frame needs to be constructed, and it is difficult when the population/sample is large. Systematic Sampling For a systematic sample, the required elements are selected at regular, chosen intervals from an ordered list. For example, if you had a population of 50 and wanted a sample of 10, use a random number generator to pick a number between one and five to find the first person, then chose every fifth after the first. Advantages include that it is simple and quick and works for large samples and populations Disadvantages include that a sampling frame is needed and, if this is not random, bias can be introduced. Stratified Sampling For stratified sampling, the population is divided into mutually exclusive strata, and a random sample is taken from each. These strata could be gender, eye colour etc. It is important that the proportion of each strata should be representative of the population, for example if 40% of a population are males and 60% female, a sample of 10 should have 4 males and 6 females. Advantages are that it reflects the population structure and gives proportional representation Disadvantages are that the population must be divided into mutually exclusive strata, and that the selection of members for each strata has the same issues as simple random sampling Non-Random Sampling There are two main types of non-random sampling: Quota Sampling Quota sampling is when a researcher selects a sample that reflects the characteristics of the whole population. Individuals are screened to see which quota they fit into, and this continues until each quota is filled. Advantages include that it allows a small sample to represent a large population, no sampling frame is needed, it is quick and easy and allows for comparison between different groups. Disadvantages include that it can introduce bias, group divisions can be vastly inaccurate, and people who do not easily fit into a group are ignored. Opportunity Sampling Opportunity sampling, also known as convenience sampling, involves taking the sample from whoever is readily available at the time and fits the criteria. For example, this might just be the first 10 people you find. Advantages are that it is extremely quick and easy Disadvantages are that it is very unlikely to represent the population and is highly dependent on the individual researcher. Location & Spread The position of something in a data set can be described using a measure of location, such as the mean, median and mode: The mode is the value that occurs most often The median is the middle value when data points are in order The mean is calculated using: Variance & Standard Deviation The variance is a measure used to describe the spread of a data set: The standard deviation is the square root of the variance: Generally, it is easiest to use the first form of the equation (without the Sxx) when you have raw data. The second one (with Sxx) is best used when you can use a calculator to find out Sxx quickly. When working with frequencies, use this equation for the variance, σ², instead: Again, standard deviation, σ, is given as the square root of this. Ranges Another form of describing the spread of a data set is using ranges. 'The' range is the difference between the largest and the smallest value Interquartile range is the difference between the upper and lower quartile, Q₃ - Q₁ Interpercentile range is the difference between the values at two given percentages Range is a good measure because it takes into account all the data, but it can be very unreliable at times, as it is affected considerably by extreme values (outliers). Interquartile range is therefore better, as it ignores extreme values and only looks at the central 50% of data. Often, the 10th to 90th percentile range is used as it also ignored outliers, but covers 80% of data rather than 50%. You can estimate percentiles and ranges by interpolation. This assumes the data is evenly distributed. When working with quartiles and percentiles, if the value you calculate for the quartile/percentile is a whole number, add a half to it. If it is a decimal, round up. Coding Statistical calculations can be simplified by coding each data value to make a new data set that is easier to work with. Where a and b are constants Box Plots An outlier is an extreme data point that does not match the trend of the other result. Generally, a value is defined as an outlier if it is some multiple, k, of the interquartile range (IQR) above or below the upper and lower quartiles respectively: A value is an outlier if it is > Q₃ + k(Q₃-Q₁) or < Q₁ - k(Q₃-Q₁) A box plot is a visual representation of a data set, and shows all the key measures clearly: Box plots are great ways of comparing different data sets: The diagram clearly shows key features for comparison: The two data sets share the same median The red set has a larger IQR The red set has fewer outliers Cumulative Frequency When the data you are given is grouped into frequencies, you need to draw a cumulative frequency diagram to estimate the median and quartiles. Always plot the upper class boundary on cumulative frequency diagrams Histograms Histograms are used to represent grouped continuous data. They are good as visual representations for data, because they show clearly where and how it is distributed. Area ∝ frequency Frequency density = frequency / category width Joining the top middle of each bar with a straight line gives the frequency polygon.
- Correlation & Regression
Bivariate data is data with two variables, and can be represented in a scatter diagram. We can describe the correlation between the two variables based on how much of a straight line the points on the diagram form. Correlation describes the nature of the linear relationship between two variables. A negative correlation occurs when one variable increases as the other decreases. A positive correlation occurs when both variables increase together. Causation The relationship can be described as causal if a change in one variable induces a change in the other. It is vital to remember that just because there may be a correlation, no matter how strong, between two variables, it does not mean the relationship is causal. Correlation does not imply causation You need to consider the context of the variables and use common sense to decide whether or not there is causation as well as correlation. Measuring Correlation The product moment coefficient, r, is a measure of strength for linear correlation between two variables. It takes values from -1 to 1, where If r = 1 the correlation is perfect and positive If r = 0 there is no correlation at all If r = -1 the correlation is perfect and negative You calculate the product moment coefficient using a stats-equipped scientific calculator. On a CASIO ClassWiz fx-991EX, to calculate the product moment coefficient, r: Click MENU Click 6: statistics Click 2: y=a+bx Input your data in the table Click AC Click OPTN Click 3: Regression Calc r is the product moment coefficient Linear Regression The line of best fit on a scatter diagram approximates the relationship between the variables. The most accurate form of line of best fit is the least squares regression line, which minimises the sum of the squares of the distances from each data point to the line. The regression line is plotted in the form y = a + bx Where b tells you the change in y for each unit change in x. If the correlation is positive, so is b, and vice versa. To calculate a and b, use your calculator and follow the steps above for the product moment coefficient. Independent & Dependent Variables The independent variable is the one that is being changed, the dependent variable is the one being measured and recorded. The independent variable should always be plotted on the x-axis The dependent variable should always be plotted on the y-axis You should only ever use the regression line to make predictions for the dependent variable Exponential Models Exponentials and logarithms can be used to model non-linear data that still has a clear pattern. If the equation is in the form y = axⁿ, a graph of log(y) against log(x) will give a straight line where log(a) is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log(y) against x will give a straight line where log a is the y intercept and log b the gradient.
Probability is used to predict the likeliness of something happening. It is always given between 0 and 1. An experiment is a repeatable process that can have a number of outcomes An event is one single or multiple outcomes A sample space is the set of all possible outcomes For two events, E₁ and E₂, with probabilities P₁ and P₂ respectively: To find the probability of either E₁ or E₂ happening, add the two probabilities, P = P₁ + P₂ To find the probability of both E₁ and E₂ happening, multiply the two probabilities, P = P₁ x P₂ The sample space for rolling two fair six-sided dice and adding up the numbers that show would look like this: To work out the probability of getting a particular result, you count how many times the result occurs and divide by the total number, 36 (since 6² = 36). So to work out the probability of getting a 10, count the number of tens and divide by 36: 3/36 = 0.0833 Generally, give your answers as decimals to three significant figures Conditional Probability If the probability of an event is dependent on the outcome of the previous event, it is called conditional. Conditional probability is noted using a vertical line between the events: The probability of B occurring, given that A has already occurred is given by P(B|A) For two independent events: P(A|B) = P(A|B') = P(A) Experiments with conditional probability can be calculated using a two-way table/restricted sample space: Venn Diagrams A Venn diagram is used to represent events happening. The rectangle represents the sample space, and the subsets within it represent certain events. Set notation is used to describe events within a sample space: A ∩ B represents the intersection A ∪ B represents the union Adding a dash, ', means the compliment, or "not" The addition formula is very useful: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive Events As you can see from the Venn diagrams, mutually exclusive events do not intersect. This means there is no overlap, so: P(A or B) = P(A) + P(B) Independent Events When one event has no effect on the other, the two events are described as independent. Therefore: P(A ∪ B) = P(A) x P(B) Conditional Probability in Venn Diagrams You can find conditional probability easily from Venn diagrams using the multiplication formula: P(B|A) = P(B ∩ A) / P(A) Tree Diagrams Tree diagrams are used to show the outcomes of two or more events happening, one after the other. For example, if there are 3 red tokens and 7 blue tokens in a bag, and two are chosen one after the other without replacement (the first is not put back into the bag), a tree diagram can model this: When you have worked out the probability of each branch, add them together - if they sum to 1, it is correct. Conditional Probability in tree diagrams Tree diagrams show conditional probability in their second and third etc columns. The multiplication formula still applies: P(B|A) = P(B ∩ A) / P(A)
- Statistical Distributions
A random variable is a variable whose value is dependent on the outcome of a random event. This means the value is not known until the experiment is carried out, however the probabilities of all the outcomes can be modelled with a statistical distribution. There are multiple ways of writing out a distribution: The examples above are all the distribution of a fair, six-sided die. The probability of each outcome is the same, so it is called a discrete uniform distribution. The sum of all probabilities in a distribution must equal 1 The Binomial Distribution If you repeat an experiment multiples times (each time is known as a 'trial'), you can model the number of successful trials with the random variable, X. A binomial distribution is used when: There are a fixed number of trials, n There are only two possible outcomes (success or fail) The probability, p, of success is constant All trials are independent If the random variable X is distributed binomially with n number of trials and fixed probability of success p, it is noted as: X∼B(n, p) Generally, you use a calculator to calculate binomial problems, but you can also use the probability mass function: There are two forms of the binomial distribution: exact and cumulative: Exact Binomial Problems This is when a question asks you to find the probability of there being a specific number of successes out of the number of trials, n. For example, to find the probability of there being exactly 4 successes for the random variable X∼B(12, 0.4) use an exact binomial distribution. To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 4: Binomial PD Click 2: Variable Input your values - in this example, x=4, N=12, p=0.4 Click = You should find the answer for this example is 0.213. This means there is a 21.3% chance of getting exactly four successful trials, out of a total 12 trials. Using the table function allows you to see the probabilities for multiple values of x. Cumulative Binomial Problems Generally, this is used more, and is used to find the sum of all the probabilities up to and including a certain value of x: P(X ≤ x) For example, if you want to find the probability of there being up to and including 4 successes (so there could be 0, 1, 2, 3 or 4 successes) for the random variable X∼B(12, 0.4), use the cumulative binomial distribution. This can be done in two ways: Tables There are tables with values for this, typically found at the back of formula books. These will have the most commons values for n, and some standard probabilities. Calculators To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click DOWN Click 1: Binomial CD Click 2: Variable Input your values - in this example, x=4, N=12, p=0.4 Click = Tables and calculators only ever give the probability for 'up to and including x', P(X ≤ x) Therefore, if you want other forms, such as P(X > a) you need to use the following functions: For P(X > a), use 1 - P(X ≤ a) For P(X < a), use P(X ≤ (a-1)) For P(X ≥ a), use 1 - P(X ≤ (a-1)) For P(X ≤ a), use P(X ≤ a) These rules work because the sum of all the probabilities equals 1. The Normal Distribution The normal distribution is used to model continuous random variables. These are variables that can take absolutely any value. The probability that the continuous random variable takes a particular specific value is always zero, but we can calculate the probability that it takes a value within a certain range. This is because continuous random variables have a continuous probability distribution: It is modelled as a curved graph The probability is the area under the curve The area under the curve can only be defined for ranges, as the area of an infinitely narrow line is zero Because the area under the graph is the probability, and the sum of all probabilities is 1, the area under the whole graph = 1 You can think of a continuous probability distribution as a histogram with an infinite number of infinitely narrow categories: The Normal Distribution A normal distribution is a continuous probability distribution that is bell-shaped and symmetrical about the mean. μ is the population mean, and is in the middle of the distribution σ is the standard deviation of the population σ² is the population variance The graph is symmetrical about the mean The graph has a total area of 1 There are points on inflection at μ + σ and μ - σ If the continuous random variable X is distributed normally with population mean μ and standard deviation σ, it is noted as: X∼N(μ, σ²) All things in nature tend to be modelled with a normal distribution (hence the name), especially heights and lengths of members of a population. It is good to know how the data is spread across the graph: Around 68% of all data is within one standard deviation of the mean (between the two points of inflection) Around 95% of all data is within two standard deviations of the mean Around 99.7% of all data is within three standard deviations of the mean Example In the example above, the median is 180 cm, and the standard deviation is 16. Therefore, continuous random variable, X, is modelled: X∼N(180, 16²). Find P(170 < X < 190) To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 2: Normal CD Input your values - in this example, lower=170, upper = 190, σ=16, μ=180 Click = You should find the answer for this example is 0.468. This means there is a 46.8% chance of someone's height being between 170 cm and 190 cm. If only one boundary is specified, e.g. P(X < 190) or P(X > 170), make the other boundary a ridiculously big negative or positive number. For P(X < 190), the upper boundary is 190, and make the lower one -9999999999999999 or something similar For P(X > 170), the lower boundary is 170 and make the upper one 9999999999999999 or something similar Inverse Normal You can also use the normal distribution backwards, to find limits from probabilities. This is done using the inverse normal distribution function. Continuing with the example above, where X∼N(180, 16²): find the value of a for which P(X < a) = 0.35. It is sometimes useful to represent this visually: To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 3: Inverse Normal Input your values - in this example, area=0.35 σ=16, μ=180 Click = You should find the answer for this example is 174. This means there is a 35% chance of someone's height being between less than 174 cm Calculators only ever calculate the area to the left This means that if you want to find the value of a for which P(X > a) = 0.35, you need to input 0.65 (1-0.35) into your calculator. Standard Normal Normally distributed variables can be standardised using coding: The standard normal distribution has mean 0 and standard deviation 1 If X∼N(μ, σ²), it can be coded into Z∼N(0, 1²) using the equation Z = (X-μ) / σ Sometimes, the probability P(Z < a) is written as Φ(a) You use your calculator normally, just enter μ = 0, σ = 1. Finding μ and σ Often, you will not know either the mean or the standard deviation of a normal distribution and will have to find it. You will, however, be given a probability, so you can code it into the standard normal distribution and solve. For example, the random variable X ∼ N(μ, 3²). Given that P(X< 10) = 0.3, find the mean. Use the inverse normal to find the value for Z when p = 0.3: Z = -0.524, so rearrange to find μ (you know X=10 and σ = 3) (3)(-0.524) = 10 - μ μ = 13.572 = 13.6 If you know neither the mean nor standard deviation, but have two probabilities, set up simultaneous equations and solve these. Approximating a Normal Distribution Binomial distributions become difficult to work with when n is large. In these instances, if p is close to 0.5, the model can be approximated with a normal distribution: If n is large and p ≈ 0.5, then X∼B(n, p) can be approximated as Y∼N(μ, σ²) where μ = np and σ² =np(1-p) Remember to square root the variance The binomial distribution is discrete, whereas the normal distribution is continuous. This means you need to apply the continuity correction whenever you approximate a binomial normally. This means that you need to add or subtract 0.5 to account for rounding: P(X < a) ≈ P(Y < a+0.5) P(X ≤ a) ≈ P(Y < a+0.5) P(X = a) ≈ P(a-0.5 < Y < a+0.5) P(X ≥ a) ≈ P(Y > a+0.5) P(X > a) ≈ P(Y > a+0.5)
- Hypothesis Testing
A hypothesis is a statement that has yet to be proved. In statistics, the hypothesis is about the value of a population parameter, and can be tested by carrying out an experiment or taking a sample of the population. The test statistic is the result of the experiment / the statistic generated from the sample. In order to perform a hypothesis test, two hypotheses are required: The null hypothesis, H₀ is the one you assume to be correct The alternative hypothesis, H₁ is the one you are testing for, to see if the assumed parameter is correct or not. A specific threshold for the probability of the test statistic must also be defined. If the probability of the test statistic is lower than this threshold, there is sufficient evidence to reject H₀. If it is above the threshold, there is insufficient evidence to reject H₀. This threshold is known as the significance level, and is typically set at 1, 5 or 10%. When ending a hypothesis test, you must conclude by saying whether or not there is sufficient evidence to reject H₀. Do not say accept or reject H₁ Critical Regions & Values If the test statistic falls within the critical region, there is sufficient evidence to reject H₀. The critical value is the first value to fall inside the critical region. The acceptance region is the set of values that are not in the critical region, so there is insufficient evidence to reject H₀. The actual significance level is the probability of incorrectly rejecting the null hypothesis. What this actually means is that: the actual significance level is the probability of getting the critical value One- and Two-Tailed Tests Hypothesis tests can be one-tailed or two-tailed. This refers to how many critical regions there are: For a one-tailed test, H₁: p < ... or H₁: p > ... and there is only one critical region For a two-tailed test, H₁: p ≠ ... so there are two critical regions, one on each 'tail' See the examples below. Hypothesis Tests on Binomial Distributions Often, hypothesis tests are carried out on discrete random variables that are modelled with a binomial distribution. One-Tailed Example A discrete random variable, X, is distributed as B(12, p). Officially, X is distributed with a probability of 0.45. However, there is a suspicion that the probability is, in fact, higher. Find, at the 5% significance level, the critical region and actual significance level of the hypothesis test that should be carried out. Write out the hypotheses & test statistic H₀: p = 0.45 H₁: p > 0.45 X∼B(12, p) Since we are only looking at whether or not the probability is more than 0.45, it is a one-tailed test. Therefore, look for the first value of X for which the cumulative probability is more than 0.95 (1 - 0.05, the 5% significance level) As you can see, the first value to have a cumulative probability of more than 0.95 is 8, so: The critical value is 8 The critical region is > 7 Find the actual significance level 1 - 0.964 = 0.036 0.036 = 3.6 % The actual significance level is 3.6% Write a conclusion If the experiment were repeated 12 times, and 8 or more of the 12 trials were successful, there would be sufficient evidence to reject H₀, suggesting the probability is indeed higher than 0.45 Two-Tailed Example a. A manufacturer of kebab-makers (a kebab-maker-maker, if you will) claims that just 25% of the kebab-makers he makes make low quality kebabs. At the 10% significance level, find the critical region for a test of whether or not the kebab-maker-maker's claim is true for a sample of 12 kebab-makers. Write out the hypotheses and test statistic H₀: p = 0.25 H₁: p ≠ 0.25 X∼B(12, p) We do not know if the probability could be more or less than 0.25, so the test is two tailed. Therefore, divide the significance level by two, and find the critical region. This will be any cumulative probability that is less than 0.05 or more than 0.95 Here you can see the critical region is in two parts, one at each 'tail' of the values; The critical region is X < 1, X > 5 b. A random sample of 12 kebab-makers is taken, and 5 are found to make low quality kebabs. Does this imply the kebab-maker-maker is lying? Method 1 See if 5 is in the critical region 5 is not > 5 not < 1 Conclude 5 does not lie within the critical region for this test (X < 1, X > 5), so there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Method 2 Find the cumulative binomial probability when X=5 When X∼B(12, 0.25), P(X=5) = 0.946 Conclude P(X=5) = 0.945, which is not within the significance level for the test. Therefore, there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Hypothesis Tests on Normal Distributions You can carry out hypothesis tests on the mean of a normally distributed random variable by looking at the mean of a random sample taken from the overall population. To find the critical region or critical value, you need to standardise the test statistic: Then, you can use the percentage points table to determine critical regions and values, or you can use the inverse normal distribution function on a scientific calculator. Example The kebabs that the kebab-maker makes have diameter D, where D is normally distributed with a mean of 4.80 cm. The kebab-maker is cleaned, and afterwards a 50 kebabs are made and measured, to see if D has changed as a result of the cleaning. D is still normally distributed with standard deviation 0.250 cm. Find, at the 5% significance level, the critical region for the test. Write out your hypotheses H₀: μ = 4.8 H₁: μ ≠ 4.8 Assume H₀ is true: Sample mean of D, Ď ∼ N(4.8, 0.25²/50 ) Code data: Z = (Ď - 0.48) / (0.25/√50) Z ∼ N(0, 1) The test is two tailed, so area on each side should be 0.025 (half of 5%): Decode, using ±1.96 (Ď - 0.48) / (0.25/√50) = -1.96 Ď - 0.48 = -0.0693 Ď = 0.411 (Ď - 0.48) / (0.25/√50) = 1.96 Ď - 0.48 = 0.0693 Ď = 0.549 Conclude The critical region is when the sample mean is smaller than 0.411 or larger than 0.549 Hypothesis Tests for Zero Correlation You can determine whether or not the product moment coefficient, p, of a sample indicates whether or not there is likely to be a linear relationship for the wider population using a hypothesis test. Use a one-tailed test if you want to test if the population p is either > 0 or < 0 Use a two tailed test if you want to see that there is any sort of relationship, so p ≠ 0 The critical region can be determined using a product moment coefficient table. It depends on significance level and sample size. To calculate the product moment coefficient of the sample, use your calculator (see notes sheet on regression & correlation).
- Indices & Algebraic Methods
When working with indices, there are eight laws that must be followed: These can be used to factorise and expand expressions. Surds Surds are examples of irrational numbers, meaning they do not follow a repeating pattern but go on forever, uniquely. Pi is the most common example of an irrational number, but surds are slightly different - they are the square roots of non-square numbers. √4 = 2 4 is a square number, so gives a rational square root √2 = 1.4142... 2 is not a square number, so its square root is irrational Like with indices, there are rules that apply to surds: These can be used to rationalise denominators: For fractions in the form 1 / √a, multiply both numerator and denominator by √a For fractions in the form 1 / (a + √b), multiply both numerator and denominator by (a - √b) For fractions in the from 1 / (a - √b), multiply both numerator and denominator by (a + √b) This is known as the conjugate pair (switching the sign of the denominator) Algebraic Fractions To simplify algebraic fractions, factorise whatever can be factorised so that parts of the numerator and denominator can cancel: Multiplication To multiply fractions, any common factors can be cancelled before multiplying the numerators and denominators. Division To divide fractions, multiply the first fraction by the reciprocal of the second fraction (flip the second fraction). Addition & Subtraction To add or subtract one fraction from another, a common denominator must be found. Long Division of Polynomials Polynomials are expressions that contain only rational numbers, positive indices and/or variables in the numerators. 3x + 5 and 3x² + 5x + 7 are examples of polynomials 3/x, √x and 5x-² are not polynomials Polynomials can be divided by (x ± p), where p is a constant, using long division: The Factor Theorem The example above divides perfectly - it does not have a remainder. This means that (2x+1) must be a factor of the initial expression we divided it into. There is a quicker way to check this: the factor theorem. The factor theorem states that if f(x) is a polynomial, then: if f(p) = 0, then (x-p) is a factor of f(x) if (x-p) is a factor of f(x), then f(p) = 0 The Remainder Theorem This can be used to find the remainder of a long division, without actually doing the division. If (x-a) is not a factor of f(x), then the remainder is given as f(a) Partial Fractions If a fraction has two or more distinct factors in its denominator, it can be separated into partial fractions. Two Linear Factors Three Linear Factors This method cannot be used if two of the factors are the same (repeated) Repeated Factors Improper Partial Fractions An improper algebraic fraction is a fraction where the numerator has an equal or higher power to the denominator. These must first be converted into proper fractions before they can be expressed as partial fractions. There are two methods of doing this: Use algebraic long division and add the remainder divided by the divisor to the quotient Multiply by the divisor and add the remainder You can find the remainder using the remainder theorem. Mathematical Proof A mathematical proof is a logical argument to show that a conjecture (a mathematical statement) is always true. Typically, a proof begins with a theorem (a pre-established fact). There are a number of requirements for a valid mathematical proof: All information and assumptions being used must be stated Every step must be shown explicitly Every step must lead on logically from the previous step All cases must be covered A proof must always end in a statement of proof. To prove an identity, like (a+b)(a-b) ≡ a² - b², you start with the expression on one side, and manipulate it algebraically until it is exactly the same as the other side. Again, the requirements above apply. Proof by Deduction Proof by deduction means starting from a fact or definition and using logical steps to prove the conjecture. Conjecture: The product of two odd numbers is also odd Proof: If a and b are integers, then 2a+1 and 2b+1 are definitely odd integers (2a+1)(2b+1) = 4ab + 2a + 2b +1 = 2(2ab+a+b) + 1 This is in the same form as 2m+1, the standard form for an odd number Therefore, the product of two odd numbers is always odd Proof by Exhaustion This involves breaking a proof into smaller proofs and dealing with these all individually. Since it requires every possible instance to be calculated, it is impossible for an infinite range (such as the example above) but can only be used on smaller scales between set limits. Conjecture: The sum of perfect cubes between zero and 100 is a multiple of 10. Proof: The only perfect cubes between zero and 100 are 1, 8, 27 and 64 1 + 8 + 27 + 64 = 100 100 = 10(10) Therefore, the sum of cubes between zero and 100 is a multiple of 10. Proof by Counter-Example Perhaps the simplest form of mathematical proof (though often the most frustrating), proof by counter-example works by finding a single occurrence when the conjecture is false. If the conjecture is false once, it is always false. Conjecture: All even multiples of five are also multiples of 4 Counter-Example: 5x2 = 10 10 is even 10 is not divisible by 4 Therefore, not all even multiples of five are also multiples of four Proof by Contradiction Proof by contradiction works by first assuming the conjecture is untrue. Then, you show through logical steps that this is impossible, and so you conclude that the initial conjecture is in fact true. The contradiction can either be with the initial assumption, or something else that is known to be true. Conjecture: √2 is irrational Assumption: √2 is not irrational Proof by contradiction: Rational numbers can be expressed in the form a/b, where a and b have no common factors So, √2 can be written as a/b: √2 = a/b Squaring both sides gives 2 = a²/b² This can be rearranged to give a² = 2b² This means that a² must be even, and so a is also even, and can be expressed as 2n (where n is an integer) Therefore, (2n)² = 2b² This equals 4n² = 2b², which cancels to 2n² = b² This shows that b² is also even, and so is b If a and b are both even, they have a common factor, 2 This contradicts the statement that a and b have no common factors, so √2 must be irrational
- Quadratic & Simultaneous Equations and Inequalities
The standard format of a quadratic expression is ax² + bx + c There are three ways of solving quadratic equations: Factorising Factorise a quadratic in the form ax² + bx + c = 0, and set each bracket to equal 0 to find the values of x (the roots). The Quadratic Equation The (b² - 4ac) inside the square root is known as the discriminant, and is used to show how many roots a quadratic has: b² - 4ac > 0: The quadratic has two distinct roots b² - 4ac = 0: The quadratic has one repeated root b² - 4ac < 0: The quadratic has no real roots Completing the Square More commonly, quadratics are in their standard form. In this case, this version is used: Simultaneous Equations Linear Simultaneous Equations There are two ways of solving linear simultaneous equations: elimination and substitution. For example, solve the following simultaneous equations: x + 3y = 11 4x - 5y = 10 Elimination Multiply the first equation by 4 4x + 12y = 44 4x - 5y = 10 Subtract 17y = 34 y = 2 Substitute this into equation 1 x + 6 = 11 x = 5 Substitution Rearrange the first equation to make x the subject x = 11 - 3y Substitute this into the second equation and solve 4(11 - 3y) - 5y = 10 44 - 12y - 5y = 10 44 - 17y = 10 -17y = -34 y = 2 Substitute this into the rearranged equation 1 x = 11 - 3(2) x = 5 Quadratic Simultaneous Equations Two simultaneous equations, one linear and one quadratic, can have up to two pairs of solutions. Don't get confused between the tow pairs! You always use the substitution method above to solve quadratic simultaneous equations - rearrange the linear equation and sub into the quadratic: Solve the simultaneous equations, x + 2y = 3, and x² +3xy = 10 Rearrange linear equation to make x the subject x = 3 - 2y Substitute into quadratic equation & solve (3 - 2y)² + 3y(3 - 2y) = 10 9 - 12y + 4y² + 9y - 6y² = 10 9 - 3y - 2y² = 10 2y² + 3y +1 = 0 (2y +1)(y +1) = 0 y = -1/2, -1 Find the corresponding x values x = 3 - 2(-1/2) x = 4 x = 3 - 2(-1) x = 5 Get the pairs together correctly: x = 4, y = -1/2 and x = 5, y = -1 Graphing Simultaneous Equations The solutions to a pair of simultaneous equations represents the intersections between their graphs. For a linear and quadratic pair of simultaneous equations, you can use the discriminant of the substituted equation (the linear equation substituted into the quadratic equation) to show whether or not there are any solutions, and if so, how many. Inequalities There is certain notation for inequalities on a number line: Linear inequalities are rearranged to make the variable the subject. To solve a quadratic inequality: rearrange so that to the right of the inequality sign is 0 solve the remaining quadratic on the left Sketch this equation roughly to see where the roots are and if it is positive or negative Identify the correct section. Regions on Graphs It is possible to show regions closed off by one or multiple lines on a graph. Again, there is certain notation to be aware of: Dashed lines do not include the curve Solid lines do include the curve Shaded areas represent the defined region
- Sequences & Series
A sequence is a list of numbers with a particular relation; a series is the sum of such a list of numbers. Sequences are sometimes a,so referred to as progressions. Arithmetic Arithmetic Sequences An arithmetic sequence has a constant defined distance between terms, e.g. 1, 3, 5 7, 9 etc. The first term is 1 and the common difference is +2. The common difference can be positive (the sequence is increasing) or negative (the series is decreasing). To calculate the nth term, u(n), of an arithmetic sequence: where a is the first term and d the common difference. Arithmetic Series An arithmetic series is the sum of all numbers in an arithmetic sequence. The sum of the first n terms is given by: where a is the first term, d the common difference, and l the last term. Geometric Geometric Sequences A geometric sequence is a sequence where there is a common ratio, not a common difference. This means the relationship between the numbers is a multiplication, not addition/subtraction. For example, the sequence 2, 4, 8, 16, 32 etc. is geometric - each term is multiplied by 2. The formula for the nth term of a geometric sequence is: where a is the first term and r is the common ratio. Geometric Series A geometric series is the sum of the first n terms of a geometric sequence. The sum of the first n terms is given by: where a is the first term and r the common ratio. The common ration cannot be 1, else each number would be the same and the sum is just n x a. Sum to Infinity As n tends towards infinity, the sum of the series is called the sum to infinity. If a series is getting bigger, its sum tends to infinity, e.g. the series 2 + 4 + 8 + 16 + 32 etc. This happens when r > 1, and the series is known as divergent. If a series is getting smaller, its sum tends to a finite value, e.g. the series 2 + 1 + 1/2 + 1/4 + 1/8 etc. This happens when r < 1, and the series is known as convergent. When a series is convergent, we can calculate the fixed value that its sum tends to, its sum to infinity: Sigma Notation The Greek capital letter sigma, ∑, is used to note sums. Limits are shown above and below the ∑ to tell you from which term to which term to sum, followed by an expression. This is the function used to calculate every term in the sequence: There are standard results for this you can substitute into other series: If the series is given in the form of an expression, but there are too many terms to write out, just write out the first few to find the first term and common difference/ratio. Then use standard arithmetic/geometric sum equations. Recurrence Relations A recurrence relation defines each term of a sequence as a function of the previous term. This means you need to know at least one term in a sequence to work forwards or backwards from. A recurrence relation is noted as: An example of how to calculate a sequence from a recurrence relation is: There are three forms a sequence can take, depending on the recurrence relation: A sequence is increasing when u(n+1) > u(n) A sequence is decreasing when u(n+1) < u(n) A sequence is periodic if the terms repeat in a regular cycle: u(n+k) = u(n) where k is the order of the sequence. A sequence can also take none of these three forms. 1, 3, 5, 7, 9... is increasing 6, 4, 2, 0, -2... is decreasing 1, 3, 5, 1, 3, 5, 1, 3, 5... is periodic with an order of 3 1, 7, 4, -9, 12... is none of the above.
- Graphs, Functions & Transformations
When sketching graphs, it is important to clearly show and label any coordinate-axis intercepts (y-intercepts and roots) as well as any stationary points (e.g. turning points). Linear Graphs The general from for a linear graph is y = mx + c, where m is the gradient and c the y-intercept. Gradient is found as rise/run: This equation can be rearranged to give an alternate equation for a line, which is more useful when you know two points and need to know the line connecting them. y2 - y1 = m(x2 - x1) To find the length of a section of line, use Pythagoras' Theorem. Two parallel lines have an equal gradient, so will never meet. Two perpendicular lines have gradients that are each other's negative reciprocal, and so they do cross. This means that the product of their two gradients equals -1 Quadratic Graphs The general form of a quadratic expression is ax² + bx + c. All quadratic graphs are parabola-shaped, symmetrical about one turning point (this can be a maximum or minimum): For quadratics in the form ax² + bx + c, c is the y-intercept. Completing the square gives the coordinates of the turning point: When f(x) = a(x + p)² + q, the turning point is at (-p, q) The discriminant tells you how many roots there are, so how many times the graph crosses the x-axis. Cubic Graphs The general form for a cubic expression is ax³ + bx² + cx + d, and can intercept the x-axis at 1,2 or 3 points. If you do not know the coefficient, then you can find out which way the graph goes by seeing what happens as x tends to ±∞: If as x → ∞, y → ∞ and x → -∞, y → -∞, the graph is positive If as x → ∞, y → -∞ and x → -∞, y → ∞, the graph is negative Cubic graphs can have just 1 or 3 distinct roots, 1 distinct root with a repeated root, or 1 triple repeated root. A triple repeated root occurs when the graph has just one stationary point, and this is on the x-axis A distinct root occurs when the graph crosses the x-axis A repeated root occurs when the graph touches the x-axis but does not cross it To sketch a cubic, you need to know the roots. If it is given in the form ax³ + bx² + cx + d, you need to factorise it first. This will tell you how many roots it has, and where they are. Then, testing to see what happens as x tends to ±∞ shows the shape. Quartic Graphs The standard form for a quartic function is ax⁴ + bx³ + cx² + dx + e where a, b, c, d and e are real numbers and a is not zero. Again, you need to know the roots of the function and the y-intercept to be able to sketch it. Reciprocal Graphs To sketch graphs of reciprocals, such as y = 1/x, y = 1/x², or -3/x, you need to know the asymptotes. These are lines that the graphs tend towards, but never touch or cross. Graphs in the form y = k/x or y = k/x² have asymptotes at x=0 and y=0 The greater the value of the numerator, the further the graph is from the coordinate axis. The asymptotes are still y=0 and x=0, however. Functions In maths, functions are relationships that map a value from a set of inputs to a single output. The set of inputs is known as the domain, and the set of possible outputs is the range. The roots of a function are the values of x for which f(x) = 0. There are two types of functions, one-to-one and many-to-one. Anything else is not a function: Composite Functions Two functions can be combined to form a composite function: fg(x) = f(g(x)) Apply g first, then apply f to this Piece-wise Defined Functions Often, functions will be split up into two or more parts, each of which applies for a certain range of values. Modulus The modulus of a number is its non-negative (or absolute) numerical value. For example |-3| = 3. The modulus of a function, therefore, is function where all input values give a positive output, regardless of whether or not the input (or x-value) is positive or negative: For a modulus functions y = |f(x)|: When f(x) ≥ 0, |f(x)| = f(x) When f(x) < 0, |f(x)| = -f(x) This is easiest shown on a graph of y=x: However, it is also possible to have the function of a modulus, rather than the modulus of a function. This is noted as y = f(|x|), not y = |f(x)|, and represents a reflection in the y-axis: It is important not to get confused between y = |f(x)| and y = f(|x|) Inverse Functions The inverse of a function, f‾¹(x) does the exact opposite to the original function, f(x) - it maps the range of the original function to its domain. Since functions cannot be one-to-many, inverse functions can only be one-to-one. f(x) and f‾¹(x) are inverses of each other ff‾¹(x) = x The domain of f(x) is the range of f‾¹(x) The range of f(x) is the domain of f‾¹(x) The graphs of f(x) and f‾¹(x) are reflections of each other in the line y=x Transformations There are a number of different types of graph transformations, that move every single point on a graph by a certain amount in a certain way. Transformations can be expressed as functions, or as vectors. When multiple transformations are combined, do one after the other. It is generally helpful to sketch out each individual transformation on a separate axis to avoid getting confused.