Teacher: Angela (AMD)

Transcription

1 BHASVIC MATHS DEPARTMENT AS Mathematics Learning Pack Teacher: Angela (AMD) Name: Block: Tutor Group:

2 Welcome to AS Maths with Statistics Your Teacher is Angela Contact me using Your Lessons You will have 2 x double lessons per week on a Wednesday and a Thursday. You will have 36 lessons in which to learn the AS course then 16 lessons for revision, consolidation and exam practice after February half term. It is very important that you do not miss any of your 36 lessons. In your Wednesday lesson you will hand in your completed weekly assignment and have an assignment test. The test will consist of roughly 4 questions taken directly from the weekly assignment you ve just handed in. You are therefore expected to get 100% on all weekly assignment tests. BRING THIS PACK TO EVERY LESSON Work to be completed each week outside lessons This will seem like a lot of work to you compared with your experiences from school. You will need to be organised to meet all your deadlines. Watch the videos for each lesson and fill in the corresponding pages of the pack, see the planner at the front of this pack. This is usually about an hour s work. Complete the weekly assignment before Wednesday s lesson. This is usually about 2 to 3 hours work and you will probably need help to get it 100% completed. Expectations Attend all lessons. If you need to miss a lesson, tell me in advance or contact me as soon as possible to collect the missed work. COMPLETE your weekly assignment each week by the deadline. This means there are no questions you haven t done and no questions with answers which do not match the correct answers, which are given to you on the assignment. It is your responsibility to get help if you need it. Pass the weekly assignment test. This means getting at least 2/3 on the drill test (basic skills test) and at least 9/10 on the randomly chosen assignment questions. Try your best and believe in yourself. Help and Resources All videos and resources can be found on or you can scan the QR codes in the pack. Learning Buddy in class Facebook: Join BHASVIC Maths Drop in with the doubles students (help in room 7 or the hall every day) A2 Mentor (maybe)

3 Core 1 GCSE Topics (in a bit more depth) You should be already comfortable working with indices, surds, simultaneous equations and quadratic equations, but we often find that students need a lot of practice to be confident with these basic algebraic techniques. You need to be able to sketch graphs of quadratics, cubics and reciprocal graphs, and sketch transformations of any graph given to you. Coordinate Geometry At GCSE you are taught that y = mx + c but we don t use that form at A Level. You will learn more about how to construct the equation of a line and use the formulae for midpoint, gradient, distance and triangle area to solve problems similar to those you ve seen at GCSE. Calculus This is one of the few new topics in Core 1 compared to GCSE. Calculus is the study of instantaneous change. You will learn the basics such as gradients of curves and tangent lines (differentiating) and un-differentiating (integrating). Series & Sequences Notation You might already understand the notation we use for recurrence relations as you ve probably seen it at GCSE. Sigma notation will be new to you and you ll learn what å ( 2r -1) means. Arithmetic Series You ve almost certainly seen Arithmetic Series at GCSE (a sequence where you get from one term to the next by adding on a fixed number). At AS you learn the equation for the nth term the sum of the first n terms and solve problems involving Arithmetic Series. Usually this just turns into an exercise in simultaneous equations. 24 r=3

4 C1 Intersection of lines and curves - Simultaneous Equations Objective: Be able to use the substitution method to find the co-ordinates of the point(s) where two straight lines or a line and curve intersect Write down the easier equation Rearrange into y = or x = Sub that the harder equation Solve to find y (or x) Use the easy equation to find x (or y) Level 1 2x 3y 10 4x y 20 Little sketch of what you re finding: Level 2 2x 3y 10 4x y 2 20 Little sketch of what you re finding:

5 Write down the easier equation Rearrange into y = or x = Sub that the harder equation Solve to find y (or x) Use the easy equation to find x (or y) Level 3 2x 3y 10 4x 2 y 2 20 Little sketch of what you re finding:

6 Objectives: C1 Graphs 1 - Sketching Graphs of Cubic Functions To be able to sketch graph transformations of To be able to sketch graphs of factorised cubic functions Graphs of Cubic Functions Activity: Sketch the following graphs: a) b) Graphs of Factorised Cubic Functions Activity: Sketch the following graphs Sketching Checklist Crosses y axis when x = 0, y =? Crosses x axis when y = 0, roots at x =? Is there a repeated root? Positive or negative? c) d)

7 C1 Graphs 1 - Sketching Reciprocal Functions Objectives: To be able to sketch graph transformations of To understand the effect of an asymptote on the shape of the graph Activity: Sketch the following graphs: a) b) Sketching Transformation of Sketching Checklist Crosses y axis when x = 0, y =? Crosses x axis when y = 0, x =? Identify the equations of the asymptote Positive or negative? c) d) (to be completed in class)

8 C1 Graph 2 - Transformations Objectives: To be able to transform a graph using shifts, stretches and reflections Understand the difference between an inside and outside transformation Graph Transformations: Shifts Activity Sketch on this graph a) f (x) + 2 b) f(x + 2) Shifts Notation The plot (2.-3) moves to? Outside f(x) + 2 Which coordinate always changes f(x) 2 Inside f(x+2) f(x-2) Graph Transformations: Stretches Activity Sketch on this graph a) 2f(x) b) f(2x) Stretches Notation The plot (2.-3) moves to? Outside 2f(x) Which co-ordinate always changes Inside f(2x)

9 C1 Graph Transformations (continued) Graphs Transformations: Reflections Activity Sketch on this graph a) - f (x) b) f(-x) Shifts Notation The plot (2.-3) moves to? Outside -f(x) Which co-ordinate always changes Inside f(-x) Summary of Graph Transformations Type Outside Description Inside Description Shifts f(x) + a f(x+a) f(x) a f(x-a) Stretches af(x) f(ax) Reflections -f(x) f(-x)

10 C1 Coordinate Geometry 1 - Equation of a Line An equation of a line with a point (a, b) and a gradient m can be written as: Example: Find the equation of the straight line which passes through the points A( 4,5) and B (2,3) Activity: Pause the video. Complete the questions. Then play again to check. Find the equation of the line that passes through each of these pairs of points. 1. A(-2, 0) and B(1, 6) 2. R(-3, -5) and S(2, 5) 3. U(4, 2) and V(-2, 4) 4. D(-1, -3) and E(3, -1)

11 C1 Coordinate Geometry 2 Gradients,, Distance, Triangle Area, Midpoint Gradients Gradients of parallel lines are the Gradients of perpendicular lines. Example: Line 1 and Line 2 are perpendicular to each other. If line 1 has a gradient of 2, then line 2 has a gradient of (TIP! FLIP IT AND CHANGE THE SIGN!) Length of a line The distance between two points on a line can be calculated by using Pythagoras. Example: Find the distance between A(0, 4) and B( 2,0) Triangle Area Area of a triangle is ½ bh Always use the perpendicular height Draw a sketch to help visualise it. Example: Find the area of the triangle ABC where A(0,4), B(2,0) and C (0,3) Midpoint To find the midpoint of two points on a line add the x co-ordinates together and divide by two. Do the same with the y co-ordinates. Example: Find the co-ordinates of the midpoints of the line AB, where A(0, 4) and B( 2,0)

12 GCSE Revision:Completing The Square solving Quadratic Functions Objective: Be able to solve a quadratic function by completing the square Be able to complete the square when the co-efficient of x² is not 1. Example 1: Complete the square in order to solve Example 2 : Complete the square in order to solve

13 C1 Quadratics 1 - The Discriminant Objectives: Be able to use the discriminant to determine the type of roots of a quadratic function Be able to solve discriminant problems Quadratic Function Value of the Discriminant Corresponding graph of Quadratic function Number and Type of Roots

14 C1 Quadratics 2 - Sketching Quadratics Objectives: Be able to sketch the graphs of quadratic functions. Be able to identify the co-ordinates of turning points and where the graph crosses both the y and x axes. Sketching Checklist: Crosses y axis, when x = 0 Crosses x axis, when y = 0 Turning point? Use completed square form to find the co-ordinates

15 C1 Quadratics 3 - Quadratic Inequalities Objective: Be able to solve quadratic inequalities Checklist: When y = 0, roots? x =, x = When x = 0 y = Sketch the quadratic graph Identify and shade <0 or >0 the region required Final solution using correct inequality notation Checklist: When y = 0, roots? x =, x = When x = 0 y = Sketch the quadratic graph Identify and shade <0 or >0 the region required Final solution using correct inequality notation Class Activity: Describe the outlined region with an inequality, where x represents the range of numbers.

16 C1 Introduction to Differentiation Objectives: To understand how to differentiate a function To understand that the derivative is the gradient function of the curve and gives the gradient of the tangent to the curve at a particular point To be able to use the gradient function to solve problems involving tangents and normals. The gradient function gives the gradient of a tangent to the curve at a specific point on the curve When the gradient function at a y = dy dx specific point is ZERO. This means that the tangent is a horizontal line and at that point the curve has a

17 C1 Differentiation 2 Tangents/Normals Objectives: To be able to calculate the gradient and equation of a tangent to a curve at a particular point and the gradient and equation of the associated normal A NORMAL is to its tangent at a particular point on a curve. On the graph draw the tangent to the curve at the point (x1, y1). Now draw on the normal to the curve at the point (x1,y1) If the gradient function for this curve, gave the gradient of the tangent as 3. What would the gradient of the normal be? What is the difference between the gradient on a straight line and a gradient measured on a curve?

18 C1 Arithmetic Progressions Objectives: To be able to understand the notation of arithmetic progressions To be able to solve problems involving n th term To be able to solve problems using the Sum to n terms formula rog Finding the n th term of an Arithmetic Progression In general : a + (a + d) + (a + 2d) +... a + ( n-1)d Example: n th term a =, d =

19 C1 Recurrence Relations Objectives: To understand the notation of recurrence relations To be able to solve problems using recurrence relations C1 Proof of Sum to n terms formula To be able to recall the proof of the sum to n terms ormula Watch the video and write down the proof here In general : a + (a + d) a + (n-2)d + a + ( n-1)d

20 C1 Sigma Notation Objectives: To understand sigma notation To be able to solve problems using sigma notation Watch the video clip and complete the following examples: 6 2k 1 k r 2 k k 4 u n 3u k 1 un 5, u1 4 4 r 2 2 r

21 C1 Integration (Indefinate) Objectives: To be able to differentiate a function To be able to use the correct notation for integration Integration is the reverse of differentiation. If we know the gradient function we can get back to the equation of the original curve by undoing the differentiation dy If 4x 1 dx then the original curve is y = Don t you forget about me! Example 1: Don t you forget To find the particular solution you need a point on the curve in order to solve for the constant term, c dy So if 4x 1, find the equation of the original curve using the point (-1, 2) dx General Formula Examples: Don t you forget about me! Don t you forget

22 Core 2 Circles You need to know the equation for a circle and the fact that tangents to circles are perpendicular to the radius. Then you can combine this with your C1 coordinate geometry knowledge to solve problems involving circles and lines. Really fun. Calculus More differentiation, this time about finding maximum and minimum points on a graph. Also, a nice application of integration finding areas under curves. You ll learn the trapezium rule which is a formula for finding the approximate area under a curve and isn t as much fun as integrating. TOOLS First you ll learn how to work in radians. They re much better than degrees (some people think centimetres are better than inches but not everyone. EVERYONE thinks radians are better than degrees). We made up the word TOOLS, but it s a handy way to remember the 5 equations you need to solve arc of circle/triangle problems. Always check your calculator degrees or radians?! Trigonometry You already know how to solve equations like sinq = 0.5 by pressing shift sin on your calculator, but at AS you learn how to find not just one but many answers, and also how to solve more complicated equations like sin 2 q + 2cosq =1 Logarithms You know how to solve x 2 = 3but do you know how to solve 2 x = 3? No, you don t. That s why you need to learn about logarithms. There are four equations to learn which you ll then need to practice using to solve more complicated equations. Factor & Remainder Theorem You know that 3 is a factor of 12, and that 7 divided by 3 has remainder 1. In AS you learn how to understand the same statements but for algebraic expressions instead of numbers. Binomial Expansion Can you expand ( 3+ 2x) 2? What about 3+ 2x formula which allows you to expand out multiple brackets really easily. ( ) 6? The Binomial Expansion is a Geometric Series Like Arithmetic Series but instead of adding on a fixed number to get from one term to the next, you multiply by a fixed number. You ll learn the formulae for nth term and sum of n terms and also, if the fixed number is between -1 and 1, how to add up infinity many terms of the sequence.

23 C2 Differentiation: Stationary (Turning) Points and the Second Derivative ( 6 minutes) Objectives: Be able to calculate the first derivative in order to find the stationary points for a given function Be able to calculate the second derivative in order to test if the stationary points are of a maximum or minimum point on the curve.understand that stationary and turning points are synonymous The second derivative: The tester function. d2 y dx 2 This is found by differentiating the first derivative dy dx If d2 y dx2 < 0 (NEGATIVE), the turning point is a If d2 y dx2 > 0 ( POSITIVE), the turning point is a You can check the gradient of the tangent using only dy before and after dx the turning point to see how the gradient is changing in order to determine if a given point is a minimum or a maximum. If d2 y dx2 = 0, then you would need to check the change in the gradient before and after the turning point in order to see if if might be a POINT OF INFLEXION Alternatively the second derivative evaluates for a particular point if the rate of change of the gradient is positive or negative hence determining the nature of the stationary point.

24 C2 Differentiation: Optimisation Problems Objectives: Be able to write an algebraic equation to solve a problem Be able to change the subject of an algebraic equation Be able to differentiate a function and solve in order to find a minimum or maximum value Be able to use the second derivative to test a value in order to prove that it is either a maximum or minimum value. 1. Write down an equation for the surface area of the cylindrical tin 2. Make h the subject of your equation. 3 a) write down the equation for the volume of a cylinder b) substitute in your expression for h from step 2.

25 C2 Differentiation: Optimisation Problems (continued) Objectives: Be able to write an algebraic equation to solve a problem Be able to change the subject of an algebraic equation Be able to differentiate a function in order to find a minimum or maximum value Be able to use the second derivative to test a value in order to prove that it is either a maximum or minimum value. 4. You are now ready to differentiate your expression for volume with respect to the radius in order to find the value of the radius which will give the maximum volume. THINK BIG BIRD!

26 C2 Co-ordinate Geometry: Circles ( 11 mins) Obectives: Recall and use the general form of the equation of the circle to solve circle problems Be able to complete the square to find the centre and radius of a circle from an equation of a circle Use circle theorems/facts to solve problems General Equation of a circle with a centre (a,b) and radius, r How do you find the centre and radius of the circle when given the equation of a circle? Circle Facts. Copy the diagrams! Most important top tip:

27 C2 TOOLS: TOOLS Summary Page Objectives: Understand radian measure Recall and Use formula for arc length and arc sector (IN RADIANS) Solve problems with triangles, sectors and segments using TOOLS kit COMPLETE THIS PAGE WITH ALL THE FORMULA YOU NEED TO LEARN. T Triangle Area O Sector Area O Arc Length L Cosine Rule S Sine Rule

28 C2 TOOLS: Radians and Arc length(continued) Radian Measure: If the radius of the sector is equal to the arc length, then the angle of the sector measures one radian. Degrees Radians The Arc Length and Area of a Sector Formula you will learn only work if the angle is in radians.

29 C2 TOOLS: Area of a Sector(CONTINUED) C2 TOOLS: Area of a Segment Area of Segment = Area of Sector Area of Triangle Using previous formulae, in general this can be written as:

30 C2 The Sine and Cosine Rule: Sine rule

31 C2 The Sine and Cosine Rule: Cosine Rule

32 C2 Trig Identities and Equations: Trig Equations Objectives: Be able to sketch the graphs of sine, cosine and tangent functions Be able to use the graphs to solve simple trig equations Be able to change the extend the range of trig function graphs Examples: Show how to solve these trig equations. 3 sin x 0 x sin x 0 x 2 2

33 3 cos x 0 x tan x 2 0 x sinx 30 0 x 360 2

34 C2 Trig Identities and Equations: Trig Equations LEARN: 2 2 sin x cos x 1 ALSO LEARN: sin x tan x cos x sin 2 x 2sin x 3 0 sin 2 x sin x 0

35 cos 2 x sin x 1 0 sin x 4cos x 0

36 C2 Exponentials and Logarithms: Exponentials and Logarithms ( 4 minutes) How to change between log form and index form Index form B P = A Log form log B A = P 2 3 = = 9 log 5 25 = 2 log = = = 1 / = 1 log = 3 log = -1 log = 3

37 C2 Exponentials and Logarithms: Rules of Logarithms ( 7 minutes) Objectives: Learn and use the rules of logarithms to simplify and solve log equations Addition rule Example: In General: Subtraction rule Example: In General: Power rule Example: In General: Change of base rule Example: In General:

38 C2 Exponentials and Logarithms: Solving log equations ( 22 minutes) How to solve log equations of the type 2 x = 5 (index form) How to solve log equations of the type log x 5 = 7 (log form) Is it in the form B P = A? Solving easy index equations Solve 1. 2 x = x = x = x = 7

39 Solving harder index equations 7 2x 7 x 56 9 x 1 3 x 3 x 3 0

40 C2 Algebra and Functions: Factor/Remainder Theorem (11 minutes) 2x 3 x 1 x 4 2 x 3 2 x x 4 x 1 The Remainder Theorem The Factor Theorem

41 C2 Algebra And Functions: Factorising Cubics (5 minutes) Factorising Cubics 3 2 Factorise 2x 7x 14x 5 using the fact that 2x 1 is a factor

42 C2 Integration: Definite integration (13 minutes) Objectives: Be able to integrate a function between two limits using correct notation dx x x dx x dx x x

43 C2 Integration: Area Under A Curve (easy) Find the area bound by the graph y = x 2 + 2, the lines x = 1, x = 5 and the x axis Find the shaded area y 3 1 y x x C P R O A x

44 C2 Integration: Area Under A Curve (easy) (CONTINUED) Find the shaded area 1 5 y x x x y O x C

45 C2 Integration: Area Under A Curve (harder) y A R B O x 2 Find the area shaded between the curves y9 x and y x 2x 3

46 R is the region bounded by Find the area of R y x x 2 6 and y 2 x y L R C O x

47 C2 Integration: Area Under A Curve (even harder) 3 2 The curve y 2x 6x 10, x 0 passes through the point A(1, 6) and has a minimum turning point at B. y A C O B x

48 (Continued) C2 Integration: Area Under A Curve ( even harder) y A R B 3 2 y = x 8 x + 20x O N x Find the shaded region

49 C2 The Trapezium Rule Write down the formula for the trapezium rule:

50 Example Question C2 The Trapezium Rule (continued)

51 C2 Geometric Progressions: Geometric Sequences and the n th term Examples of Geometric Sequences 2, 4, 8, , 100, 10, 0.1 Each term in a geometric sequence has been multiplied by a common ratio, r to generate the next term. The first term is normally given the letter, a Finding the n th term of a geometric progression: Finding the 20 th term in the GP, Finding the number of terms in the GP,

52 C2 Geometric Progressions (Continued) Find the tenth term of the series Find the number of terms in the GP C2 Geometric Progressions: Sum of geometric progression ( 3 minutes) Proof of the sum of a geometric progression

53 C2 Binomial Expansion: Binomial Expansion b a b a b a b a b a b a Finding the coefficients Method 1: Method 2: Method 3:

54 2 3x 4 1 x 7 2 x 2 4

55 2 px 4 1 2x n

56 Statistics 1 GCSE Data Handling Revision of finding the mean, median, quartiles and standard deviation from raw data and data in frequency tables. You will also need to be able to interpret the skewness of the data by using a variety of methods. You will need to be able to interpret histograms and calculate the dimensions of bars in a histogram. Most of this should be a revision of what you have studied at higher tier GCSE. In addition you will need to understand how coding of data affects the mean and the standard deviation. Probability You will need to be able to use possibility space diagrams, two way tables and tree diagrams; this should be revision of what you have already studied. In addition we will look at Venn Diagrams and their notation and how they can be used to solve probability problems. We will also look at relationships such as mutually exclusive and independent events and rules affecting conditional probability. Discrete Random Variables We will look at the probability distributions of random variables such as rolling a dice etc. We will see how we can work out the expected value and variance of a particular event occurring. Normal Distribution We will look at the normal distribution; this shows the particular distribution of certain continuous variables such as height and time. The normal distribution can be used to estimate probabilities of the likelihood of certain events occurring. Correlation and Regression Following on from your look at Scatter Diagrams at GCSE we move it on by trying to put a numerical value on the strength of correlation that the scatter diagram suggests, using the Product Moment Correlation Coefficient. We then will look at the use of Regression to try and give the line of best fit an equation in the form of y = mx + c.

57 S1 GCSE Data Handling Median and Quartiles from a list Example: Stem and Leaf Diagram Address: Where to find the quartiles? n is number of values n Lower Quartile, Q Median, Q Upper Quartile, Q 4 Example: Copy down example of the quartiles Rounding Rules: If the address not a whole number then Address is a whole number then Inter- Quartile Range (IQR) Example: Q Q Outliers: extreme values or anomalies that unnecessarily skew the data IQR IQR. x Q. x Q Example:

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59 GCSE Data Handling Mean and Standard Deviation from a list Number of Goals Scored by Palace Mean x x n Standard Deviation Example: Example: σ Variance = Number of Goals Scored by Brighton Mean: x x n Standard Deviation: Example: Example: σ Variance = variance is the mean of the x s squared minus the mean squared Comparisons: Palace Brighton

60 GCSE Data Handling : Skew Type Positive No Skew Negative Tail Tail in higher values (on x axis) Symmetrical Tail in lower values(on x axis) Position of mean? Box Plots and quartiles Example: What s the skew GCSE Data Handling Box plots Outliers: extreme values or anomalies that unnecessarily skew the data IQR. x Q IQR. x Q

61 GCSE Data Handling This video shows you how to find the mean and standard deviation of grouped and ungrouped frequency tables. Mean and Standard Deviation from a frequency table Ungrouped Frequency Maths test mark (x) Number of people (f) Find the mean and the standard deviation Mean: x fx f Example: Standard deviation: σ Example: Grouped Frequency Height Frequency x fx x² f x² Find the mean and the standard deviation Mean: x fx f Example: Standard deviation: σ Example:

62 Data Handling Effects of Coding on Mean and Variance X, is the number of goals scored in a netball match 1,2,3,4,5 Calculate the mean and variance of 1,2,3,4,5 Calculate the mean and variance of 11,12,13,14,15e Calculate the mean and variance off 10,20,30,40,50

63 GCSE Data Handling This video shows you how to work out the median and quartiles from a frequency table called INTERPOLATION Median and Quartiles from a frequency table n = t = gw = gf = gb = median = n = t = gw = gf = gb = lower quartile = Upper Quartile n = t = gw = gf = gb = upper quartile =

64 GCSE Data Handling Median and Quartiles from a frequency table (CONTINUED) Pause the video clip and have a go yourself. Then play the video and check if you are right. GCSE Data Handling Histograms This video shows you the key features of Histograms.

65 Data Handling Histograms: How to find the dimensions of bars This video shows you how to find the dimensions of a Histogram IMPORTANT!!! CHECK THE BAR WIDTH!

66 S1 Probability - Tree diagrams This video reminds you how to use a tree diagram A bag contains 3 blue balls and 5 red balls. Two are selected at random without replacement. Find the probability that a) They are both blue b) There is one of each colour Pythagoras tree look it up! There are 5 black socks and 3 reds in a bag. I pick 2 socks without replacement. Find the probability I get a) Two of the same coloured sock b) At least one red sock

67 S1 Probability - Sample space diagrams This video introduces the idea of a sample space diagram A dice house Two dice are thrown and their scores added. Find the probability of getting a) A total of 7 b) A total of 8 or more c) A score of 5 or more given the first score was even

68 Two coins are thrown. What is the probability of getting two tails? Spinner A has 2,3,4 and spinner B has 1, 2,3,4,5. They are spun and their scores are added. Find the probability of a) A total of 6 b) A total of 7 or more c) A total of 4 or more given that spinner A was an even number

69 S1 Probability Introduction to Venn Diagrams Venn Diagrams P A B Formulae to Learn: Addition Rule OR = AND = Mutually Exclusive: Independent:

70 S1 Probability Introduction to Venn Diagrams (Continued) Example 1 A school has 97 students in year 12, 55 take Maths and 32 take Chemistry. 31 take neither. How many take both?

71 S1 Discrete random variables DRVs - Probability distributions This video explains what a DRV is and shows you how to find the probability distribution Find the probability distribution for S the score on a die. A die is thrown until either a six appears or I ve thrown it three times. Find the probability distribution for T where T is the number of throws.

72 S1 Discrete random variables DRVs Probabilities total 1 This video shows you the important fact that the total probability is 1. Not much to write here! Just remember P ( X x) 1 The sum of the probabilities = 1!!!!!!!!!

73 S1 Discrete random variables DRVs - F(X) This video shows you the cumulative distribution function F(X) Here is a probability distribution X P(X = 1/10 4/10 2/10 1/10 2/10 x) F(3) = F(2.7) = Find the cumulative distribution function X F(X) F(X) = This just means it s the sum of the probability and all the probabilities below it Can you go back the other way? Now imagine we were given a cumulative distribution table X F(X) 1/10 2/10 4/10 8/10 10/10 and asked to find the probability distribution table X P(X=x)

74 S1 Discrete random variables DRVs - E(X) This video shows you how to find the expected value or mean of a DRV. The probability distribution is: X P(X = 10/60 20/60 15/60 5/60 x) Find E(X) X P(X = 1/6 2/6 2/6 1/6 x) Find E(X)

75 S1 Discrete random variables DRVs - Var(X) This video shows you how to calculate the variance of a DRV. X P(X = 10/60 20/60 15/60 5/60 Variance squariance x) Find Var (X) X P(X = 1/6 2/6 2/6 1/6 x) Find Var (X) Did you remember to minus the mean squared?

76 S1 Normal distribution An introduction This video gives an introduction to the Normal Distribution Continuous Data such as height of an adult, time taken to run a race etc tend to have a NORMAL DISTRIBUTION. If data is known to have a NORMAL DISTRIBUTION, there are certain characteristics we can assume: It is a SYMMETRICAL distribution and is centered on the mean of the data. 99.8% of the data lies within 3 standard deviations of the mean The mean and standard deviation can be used to standardise the distribution in order to estimate probabilities of events using z values A normal distribution with a mean of 0 and a standard deviation of 1 has probabilities already calculated for it. These are the z values. How to standardise? The formula to standardise is z = Example:

77 S1 Normal distribution Standard Normal Distribution This video shows you how to use the z values for the standard normal distribution

78 S1 Correlation & Regression - Product moment correlation coefficient (pmcc) This video looks at scatter graphs and how we calculate and interpret the pmcc. Find the PMCC and comment on it. English marks Maths marks These are in the formula book

79 Find the PMCC and comment on it. Maths marks English marks

80 S1 Correlation & Regression The regression line This video shows you how to find an equation for the regression line (line of best fit) Mins spent Maths marks revising The equation of the least squares regression line y on x is y = a + bx where and Find the equation of the least squares regression line y on x Interpret a Interpret b

81 Use your line to find the Maths mark if you revise for 101 hours 110 hours Hours spent English marks revising Find the equation of the least squares regression line y on x Interpret a Interpret b Use your line to find the English mark if you revise for 6.5 hours 10 hours