2 Volume 3 of “Teach A level Maths” covers the work on Probability and Statistics for the A/AS level Option Module S1.All topics for the 4 specifications offered by the English examining bodies are covered. Where a topic relates to some specifications only, this is indicated in a contents file and also at the start of the presentation.
3 Explanation of Clip-art images An important result, example or summary that students might want to note.It would be a good idea for students to check they can use their calculators correctly to get the result shown.An exercise for students to do without help.
4 The slides that follow are samples from 9 of the 40 presentations. 23: Binomial Problems4: Box and Whisker Diagrams6: Histograms26: Hypothesis Testing10: Introduction to Probability28: Standardizing to Z14: Discrete Random Variables36: Calculating Residuals16: Linear Functions of a Discrete Random Variable
5 4: Box and Whisker Diagrams Demo version note: The S1 specifications require students to be familiar with topics covered in Data Handling at GCSE.The first few presentations revise and extend the GCSE work.By the time the students reach this 4th presentation they have been reminded about cumulative frequency diagrams and have met the age data referred to on the slide.
6 Box and Whisker Diagrams The diagram can easily be drawn using a cumulative frequency diagram.The projected population of the U.K. for 2005( by age )I’ll use the age data that we met earlier.The box can be any depth.minimum ageOne whiskerThe boxThe other whiskermedianmaximum agelower quartileupper quartile
7 Box and Whisker Diagrams The diagram can easily be drawn using a cumulative frequency diagram.The projected population of the U.K. for 2005( by age )I’ll use the age data that we met earlier.medianminimum agemaximum agelower quartileupper quartile
8 Box and Whisker Diagrams The diagram can easily be drawn using a cumulative frequency diagram.The projected population of the U.K. for 2005( by age )I’ll use the age data that we met earlier.We need a scale.10050Age (years)
9 6: Histograms Histograms Demo version note: As well as explaining theory, the presentations show worked examples and set introductory exercises.The 6th presentation reminds students about the rules for drawing Histograms. The exercise shown here reinforces these rules without the students needing to spend time drawing a diagram.
10 Histograms Exercise 0-19 20-29 30-39 40-44 45-49 50-59 60-89 5 8 16 22 95 components are tested until they fail. The table gives the times taken ( hours ) until failure.Time to failure (hours)0-1920-2930-3940-4445-4950-5960-89Number of components5816221810Find 3 things wrong with the histogram which represents the data in the table.
11 Histograms Answer: 0-19 20-29 30-39 40-44 45-49 50-59 60-89 5 8 16 22 Time to failure (hours)0-1920-2930-3940-4445-4950-5960-89Number of components5816221810Frequency has been plotted instead of frequency density.There is no title.There are no units on the x-axis.
12 Histograms Incorrect diagram Time taken for 95 components to fail
13 Introduction to Probability Demo version note: This presentation covers the introductory ideas of probability and leads to a later one on conditional probability.Summaries are given from time to time which teachers may want students to note down. This slide shows an example of a summary.
14 Introduction to Probability SUMMARYOutcomes are the results of trials or experiments.An event is a particular result or set of results.A possibility space is the set of all possible outcomes.For equally likely outcomes, the probability of an event, E, is given byP (E) =number of ways E can occurnumber of possible outcomes
15 Discrete Random Variables Demo version note: The presentations all contain worked examples of the straightforward type of questions found in exams.This is the first of the examples in the presentation on Discrete Random Variables.
16 Discrete Random Variables e.g. 1. A random variable X has the probability distributionPx(X = )pFind (a) the value of p and (b) the mean of X.Solution:(a) Since X is a discrete r.v.,(b) mean,Tip: Always check that your value of the mean lies within the range of the given values of x. Here, or 5·25, does lie between 1 and 10.
17 Linear Functions of a Discrete Random Variable Demo version note: Some topics are not required by all the specifications. The contents file shows which topics are needed by each of the specifications and contains hyperlinks to the files.The topic Linear Functions of a Discrete Random Variable is only required in S1 by Edexcel.
18 Linear Functions of a Discrete Random Variable The results we have found can be generalised to giveE(aX + b) =aE(X) + be.g. The probability distribution for the r.v. X is given by108642xFind (a) E(X), (b) Hence find E(2X - 3)Solution: (a)“Hence” in part (b) of the question means that we must use the answer to part (a) rather than using the values and probabilities of 2X - 3.(b)
19 23: Binomial Problems Binomial Problems Demo version note: The Binomial Distribution is covered by AQA, MEI/OCR and OCR.Having learnt to carry out Binomial Calculations, students practise recognising the conditions for using the model and also learn the importance of defining a random variable and writing down its distribution.
20 Are the conditions met for using the Binomial model? Binomial Problemse.g. 1. A factory produces a particular type of computer chip. Over a long period the number that are defective has been found to be 15%. What is the probability that in a sample of 20 taken at random, 19 are perfect?Are the conditions met for using the Binomial model?A trial has 2 possible outcomes, success and failure.Yes: Each chip is either defective or not.The trial is repeated n times.Yes: 20 chips are selected so n = 20.The probability of success in one trial is p and p is constant for all the trials.Yes: We are given 15% (so p = 0·15 ) and we can assume it is constant.The trials are independent.Yes: The probability of selecting a defective chip does not depend on whether one has already been selected.
21 Binomial Problemse.g. 1. A factory produces a particular type of computer chip. Over a long period the number that are defective has been found to be 15%. What is the probability that in a sample of 20 taken at random, 19 are perfect?Solution:Let X be the r.v. “number of defective chips”We must never miss out this stage since it reminds us that(i) X represents a number ( that can be 0, 1, 2, n ), and(ii) we have to make the decision as to whether to count the number of defective chips or perfect ones.So,Writing the distribution of X in this way makes us check that we have the p that fits our definition of the r.v., defective rather than perfect.
22 Binomial Problemse.g. 1. A factory produces a particular type of computer chip. Over a long period the number that are defective has been found to be 15%. What is the probability that in a sample of 20 taken at random, 19 are perfect?Solution:Let X be the r.v. “number of defective chips”So,The solution is now straightforward. We wantWe need to be very careful here and not use by mistake.I had set up the Binomial for the number of defective chips, because I had the proportion for defective. However, the question asked for the probability of 19 perfect ones.If I had writtenLet X be the r.v. “ number of perfect chips”Then,and I would want
23 26: Hypothesis Testing Hypothesis Testing Demo version note: In the presentations extensive use is made of snapshots from the software package “Autograph”.Here Autograph is used to illustrate an example on Hypothesis Testing in the presentation for the MEI/OCR specification.
24 Hypothesis Testinge.g. 2. In a trial, 16 seeds are sown and only 11 germinate. Use a 10% significance level to test the supplier’s claim that 85% germinate. Find the critical region for the test.Solution:Let X be the random variable ”the number of seeds that germinate”Test at 10% level of significance.To test the supplier’s claim, the alternative hypothesis is that fewer than 85% germinate.This is again a 1-tailed test but this time we need to test the bottom end of the distribution.There is a probability of 0·0791 ( less than 10% ) that 11 or fewer seeds will germinate.We reject the null hypothesis at the 10% level of significance and conclude that the germination rate is below 85% .
25 ( 21·01% ), so the critical region for the test is Hypothesis TestingThe Autograph illustration is as follows:The probability of 12 or fewer germinating is 0·2101( 21·01% ), so the critical region for the test is0, 1, 2, , 11.
26 Standardizing to Z28: Standardizing to ZDemo version note: Students are encouraged to use their Formulae and Statistical Tables even when worked examples are being developed.This presentation is part of a series to be used by AQA and Edexcel students on the Normal Distribution.
27 x = 400, so Standardizing to Z e.g.1 If X is a random variable with distributionfind (a) (b)Solution: (a)x = 400, soTables only give 2 d.p. for z so this is all we need.So,
28 Standardizing to Ze.g.1 If X is a random variable with distributionfind (a) (b)Solution: (b)There are 2 values to convert so we use subscripts for z.N.B. This is left of the mean so the z value will be negative.So,
29 Standardizing to Ze.g.1 If X is a random variable with distributionfind (a) (b)Solution: (b)
30 Calculating Residuals 36: Calculating Residuals Demo version note: Throughout the module, students are encouraged to use their calculators efficiently and this is particularly important in the topic for AQA, Edexcel and OCR on Least Squares Regression.In the following slides, however, the emphasis is on the effect of outliers on the equation of a regression line rather than on calculating the line itself.
31 Calculating Residuals e.g. This is a scatter diagram of the data shown in the table.xy15218312414512If we were to draw the line “by eye”, the 1st point . . .611would lie well away from the line we would want to draw.7783However, the calculation of the regression line includes the 1st point and distorts the position of the line.
32 Calculating Residuals e.g. This is a scatter diagram of the data shown in the table.3871161251441821yxThe diagram shows the y on x regression line for all the data. The residuals are shown by the red lines.The left-hand end of the line is further down than it would be without the 1st point.
33 Calculating Residuals e.g. This is a scatter diagram of the data shown in the table.3871161251441821yxRemoving the 1st point . . .
34 Calculating Residuals e.g. This is a scatter diagram of the data shown in the table.3871161251441821yxRemoving the 1st point gives
35 Calculating Residuals e.g. This is a scatter diagram of the data shown in the table.The sum of the squares of the residuals,Removing the 1st point givesThe sum of the squares of the residuals,Without the 1st point, we have a regression line that is a much better fit.
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